{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-26T00:14:02.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-04-26T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":58498,"title":"Compute the Sisyphus sequence","description":"A recent article in the New York Times featured the Online Encyclopedia of Integer Sequences, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first term is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \r\nThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \r\nAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \r\nWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, n and flag, and four outputs, an, z10, amax, and nrestart. The output an is the nth term of the sequence; z10 is a list of the 10 smallest integers not in the sequence; amax is the maximum value in the sequence; and nrestart is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\r\nIf flag is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If flag is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 372px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 186px; transform-origin: 407px 186px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA recent \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html\"\u003e\u003cspan style=\"perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; \"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003earticle in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 50.9px 8px; transform-origin: 50.9px 8px; \"\u003e\u003cspan style=\"font-style: italic; text-decoration-line: underline; \"\u003eNew York Times\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.225px 8px; transform-origin: 41.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e featured the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://oeis.org/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eOnline Encyclopedia of Integer Sequences\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.45px 8px; transform-origin: 68.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eterm\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.5083px 8px; transform-origin: 87.5083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 360.95px 8px; transform-origin: 360.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 356.7px 8px; transform-origin: 356.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 361.233px 8px; transform-origin: 361.233px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 8px; transform-origin: 3.85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3333px 8px; transform-origin: 58.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and four outputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ean\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ez10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eamax\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 8px; transform-origin: 30.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003enrestart\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.275px 8px; transform-origin: 39.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ean\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2167px 8px; transform-origin: 48.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the nth term of the sequence; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ez10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 173.858px 8px; transform-origin: 173.858px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a list of the 10 smallest integers not in the sequence; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eamax\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.858px 8px; transform-origin: 124.858px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the maximum value in the sequence; and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 8px; transform-origin: 30.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003enrestart\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 250.467px 8px; transform-origin: 250.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 362.775px 8px; transform-origin: 362.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 343.033px 8px; transform-origin: 343.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [an,z10,amax,nrestart] = Sisyphus(n,flag)\r\n  an = n+primes(n);\r\n  z10 = find(an~=1:n);\r\n  amax = max(an);\r\n  nrestart = 1;\r\nend","test_suite":"%%\r\nn = 100;\r\nan_correct = 55;\r\nz10_correct = [13 17 19 25 26 27 32 33 34 36];\r\namax_correct = 220;\r\nnrestart_correct = 3;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 100;\r\nan_correct = 110;\r\nz10_correct = [13 17 19 25 26 27 32 33 34 36];\r\namax_correct = 220;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1000;\r\nan_correct = 990;\r\nz10_correct = [25 27 36 50 54 60 72 75 79 84];\r\namax_correct = 4420;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1000;\r\nan_correct = 1980;\r\nz10_correct = [25 27 36 50 54 60 72 75 79 84];\r\namax_correct = 4420;\r\nnrestart_correct = 5;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 10000;\r\nan_correct = 24975;\r\nz10_correct = [27 36 54 60 72 79 84 97 107 108];\r\namax_correct = 59820;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 10000;\r\nan_correct = 49950;\r\nz10_correct = [27 36 54 60 72 79 84 97 107 108];\r\namax_correct = 59820;\r\nnrestart_correct = 5;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1e6;\r\nan_correct = 8820834;\r\nz10_correct = [36 72 97 107 115 127 144 167 194 211];\r\namax_correct = 9466580;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 5e6;\r\nan_correct = 8394938;\r\nz10_correct = [36 72 97 107 115 127 144 167 194 211];\r\namax_correct = 53375956;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nfiletext = fileread('Sisyphus.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-07-19T02:25:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-13T03:32:34.000Z","updated_at":"2024-12-11T00:56:19.000Z","published_at":"2023-07-13T03:32:34.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA recent \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003earticle in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNew York Times\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e featured the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOnline Encyclopedia of Integer Sequences\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eterm\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and four outputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ean\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ez10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eamax\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enrestart\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ean\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the nth term of the sequence; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ez10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a list of the 10 smallest integers not in the sequence; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eamax\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the maximum value in the sequence; and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enrestart\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":58498,"title":"Compute the Sisyphus sequence","description":"A recent article in the New York Times featured the Online Encyclopedia of Integer Sequences, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first term is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \r\nThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \r\nAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \r\nWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, n and flag, and four outputs, an, z10, amax, and nrestart. The output an is the nth term of the sequence; z10 is a list of the 10 smallest integers not in the sequence; amax is the maximum value in the sequence; and nrestart is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\r\nIf flag is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If flag is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 372px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 186px; transform-origin: 407px 186px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA recent \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html\"\u003e\u003cspan style=\"perspective-origin: 39.675px 8px; transform-origin: 39.675px 8px; \"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003earticle in the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"perspective-origin: 50.9px 8px; transform-origin: 50.9px 8px; \"\u003e\u003cspan style=\"font-style: italic; text-decoration-line: underline; \"\u003eNew York Times\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 41.225px 8px; transform-origin: 41.225px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e featured the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://oeis.org/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eOnline Encyclopedia of Integer Sequences\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.45px 8px; transform-origin: 68.45px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 8px; transform-origin: 14px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eterm\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 87.5083px 8px; transform-origin: 87.5083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 360.95px 8px; transform-origin: 360.95px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 356.7px 8px; transform-origin: 356.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 361.233px 8px; transform-origin: 361.233px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 8px; transform-origin: 3.85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003en\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58.3333px 8px; transform-origin: 58.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and four outputs, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ean\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ez10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.88333px 8px; transform-origin: 3.88333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eamax\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 17.5px 8px; transform-origin: 17.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 8px; transform-origin: 30.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003enrestart\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.275px 8px; transform-origin: 39.275px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The output \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 8px; transform-origin: 7.7px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ean\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.2167px 8px; transform-origin: 48.2167px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the nth term of the sequence; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.55px 8px; transform-origin: 11.55px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003ez10\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 173.858px 8px; transform-origin: 173.858px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a list of the 10 smallest integers not in the sequence; \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eamax\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 124.858px 8px; transform-origin: 124.858px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the maximum value in the sequence; and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.8px 8px; transform-origin: 30.8px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003enrestart\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 250.467px 8px; transform-origin: 250.467px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.825px 8px; transform-origin: 5.825px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 362.775px 8px; transform-origin: 362.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 8px; transform-origin: 15.4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; \"\u003eflag\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 343.033px 8px; transform-origin: 343.033px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [an,z10,amax,nrestart] = Sisyphus(n,flag)\r\n  an = n+primes(n);\r\n  z10 = find(an~=1:n);\r\n  amax = max(an);\r\n  nrestart = 1;\r\nend","test_suite":"%%\r\nn = 100;\r\nan_correct = 55;\r\nz10_correct = [13 17 19 25 26 27 32 33 34 36];\r\namax_correct = 220;\r\nnrestart_correct = 3;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 100;\r\nan_correct = 110;\r\nz10_correct = [13 17 19 25 26 27 32 33 34 36];\r\namax_correct = 220;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1000;\r\nan_correct = 990;\r\nz10_correct = [25 27 36 50 54 60 72 75 79 84];\r\namax_correct = 4420;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1000;\r\nan_correct = 1980;\r\nz10_correct = [25 27 36 50 54 60 72 75 79 84];\r\namax_correct = 4420;\r\nnrestart_correct = 5;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 10000;\r\nan_correct = 24975;\r\nz10_correct = [27 36 54 60 72 79 84 97 107 108];\r\namax_correct = 59820;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 10000;\r\nan_correct = 49950;\r\nz10_correct = [27 36 54 60 72 79 84 97 107 108];\r\namax_correct = 59820;\r\nnrestart_correct = 5;\r\n[an,z10,amax,nrestart] = Sisyphus(n,false);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 1e6;\r\nan_correct = 8820834;\r\nz10_correct = [36 72 97 107 115 127 144 167 194 211];\r\namax_correct = 9466580;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nn = 5e6;\r\nan_correct = 8394938;\r\nz10_correct = [36 72 97 107 115 127 144 167 194 211];\r\namax_correct = 53375956;\r\nnrestart_correct = 4;\r\n[an,z10,amax,nrestart] = Sisyphus(n);\r\nassert(isequal(an,an_correct))\r\nassert(isequal(z10,z10_correct))\r\nassert(isequal(amax,amax_correct))\r\nassert(isequal(nrestart,nrestart_correct))\r\n\r\n%%\r\nfiletext = fileread('Sisyphus.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'assert'); \r\nassert(~illegal)","published":true,"deleted":false,"likes_count":0,"comments_count":5,"created_by":46909,"edited_by":46909,"edited_at":"2023-07-19T02:25:08.000Z","deleted_by":null,"deleted_at":null,"solvers_count":6,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-07-13T03:32:34.000Z","updated_at":"2024-12-11T00:56:19.000Z","published_at":"2023-07-13T03:32:34.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA recent \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.nytimes.com/2023/05/21/science/math-puzzles-integer-sequences.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003earticle in the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNew York Times\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e featured the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://oeis.org/\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eOnline Encyclopedia of Integer Sequences\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, founded by Neil J.A. Sloane. One of the sequences discussed in the article is the Sisyphus sequence. The first \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eterm\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is 1. Subsequent terms are computed with this rule: if the previous term is even, then divide it by 2; if the previous term is odd, add the next available prime. Therefore, the sequence starts 1, 3, 6, 3, 8, 4, 2, 1, 8, 4, 2, 1, 12,… \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThe sequence gets its name from the story of Sisyphus, whom one of the Greek gods punished by making him roll a boulder up a hill. Every time Sisyphus neared the top, the boulder would roll back down—just as the sequence “rolls” down to 1 when it hits a power of 2. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eAn open question is whether every integer appears in this sequence. Some appear multiple times, but others resist appearance for quite a while. For example, 36 appears some time after a billion terms. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to generate the Sisyphus sequence and a variant and determine the smallest missing numbers. The function should have two inputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and four outputs, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ean\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ez10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eamax\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e, and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enrestart\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e. The output \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ean\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the nth term of the sequence; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003ez10\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a list of the 10 smallest integers not in the sequence; \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eamax\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the maximum value in the sequence; and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enrestart\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is the number of times the sequence rolls back to 1 (i.e., not counting the first 1).\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is true or unspecified, generate the sequence as described above—i.e., by adding the next available prime to an odd term. If \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eflag\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is false, add the next largest unused prime to an odd term. In the latter case, the sequence would start 1, 3, 8, 4, 2, 1, 4, 2, 1, 8, 4, 2, 1, 12, 6, 3, 16, 8, 4, 2, 1… 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