{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-04-16T00:12:35.000Z","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":"2026-04-16T00:00:00.000Z","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":53810,"title":"List the Beatriz numbers","description":"Write a function to find integers  less than or equal to an input value  that solve the equation . The function  gives the sum of divisors of ; for example, . The totient function  counts numbers less than  that are relatively prime to . For example,  because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 1. \r\nAn oblique hint is that I call solutions to  “Beatriz numbers” in honor of a good friend and former co-worker of EmilyR and JessicaR. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.  ","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: 156px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 78px; transform-origin: 407.5px 78px; 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: 384.5px 42px; text-align: left; transform-origin: 384.5px 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: 98.275px 7.66667px; transform-origin: 98.275px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to find integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\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: 112.417px 7.66667px; transform-origin: 112.417px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e less than or equal to an input value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\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: 74.2917px 7.66667px; transform-origin: 74.2917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that solve the equation \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n-1) = phi(n+1)\" style=\"width: 127px; height: 18.5px;\" width=\"127\" height=\"18.5\"\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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(x)\" style=\"width: 30.5px; height: 18.5px;\" width=\"30.5\" height=\"18.5\"\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: 31.8917px 7.66667px; transform-origin: 31.8917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003esum of divisors\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: 9.71667px 7.66667px; transform-origin: 9.71667px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 44.3333px 7.66667px; transform-origin: 44.3333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(15) = 1+3+5+15 = 24\" style=\"width: 178.5px; height: 18.5px;\" width=\"178.5\" height=\"18.5\"\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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003etotient function\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: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(x)\" style=\"width: 32px; height: 18.5px;\" width=\"32\" height=\"18.5\"\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: 24.5px 7.66667px; transform-origin: 24.5px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 84.4px 7.66667px; transform-origin: 84.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 46.6667px 7.66667px; transform-origin: 46.6667px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"phi(15) = 8\" style=\"width: 66px; height: 18.5px;\" width=\"66\" height=\"18.5\"\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: 127.433px 7.66667px; transform-origin: 127.433px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 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: 122.525px 7.66667px; transform-origin: 122.525px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn oblique hint is that I call solutions to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n-1) = phi(n+1)\" style=\"width: 127px; height: 18.5px;\" width=\"127\" height=\"18.5\"\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: 182.817px 7.66667px; transform-origin: 182.817px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e “Beatriz numbers” in honor of a good friend and former co-worker of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/players/18927291\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEmilyR\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: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/players/8608872\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eJessicaR\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: 268.4px 7.66667px; transform-origin: 268.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.\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 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Beatriz(m)\r\n  y = f(m);\r\nend","test_suite":"%%\r\nm = 30;\r\ny_correct = [4 6 12 18 30];\r\nassert(isequal(Beatriz(m),y_correct))\r\n\r\n%% \r\nm = 200;\r\ny_correct = [4 6 12 18 30 42 60 72 102 108 138 150 180 192 198];\r\nassert(isequal(Beatriz(m),y_correct))\r\n\r\n%% \r\nm = 1000;\r\ny = Beatriz(m);\r\nhist_correct = [8 7 4 2 3 2 4 0 5 0];\r\ns_correct = 12118;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 10000;\r\ny = Beatriz(m);\r\nhist_correct = [35 26 20 22 23 17 19 13 14 16];\r\ns_correct = 863206;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 100000;\r\ny = Beatriz(m);\r\nhist_correct = [205 137 125 124 114 106 94 102 109 108];\r\ns_correct = 54557284;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 1e6;\r\ny = Beatriz(m);\r\nhist_correct = [1224 936 834 810 761 766 730 705 706 697];\r\ns_correct = 3712183324;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 1e7;\r\ny = Beatriz(m);\r\na = y(1:2949:end);\r\na_correct = [4 294180 675840 1093638 1527678 1980660 2458368 2954772 3442122 3941688 4461642 4994790 5535240 6076308 6624972 7185888 7734828 8288340 8852070 9423180];\r\nassert(isequal(a,a_correct))\r\n\r\n%%\r\ny = Beatriz(1e9);\r\ns = sum(1./(y-1)+1./(y+1));\r\nB2 = 1.902160583104;\r\ner_correct = 0.127424625465465;\r\nassert(B2-s-er_correct\u003c1e-9)\r\n\r\n%%\r\nfiletext = fileread('Beatriz.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'oeis'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-01-10T05:12:56.000Z","updated_at":"2025-12-07T06:51:03.000Z","published_at":"2022-01-10T05:18:11.000Z","restored_at":null,"restored_by":null,"spam":false,"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\u003eWrite a function to find integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e less than or equal to an input value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that solve the equation \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n-1) = phi(n+1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n-1) = \\\\phi(n+1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum of divisors\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(15) = 1+3+5+15 = 24\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(15) = 1+3+5+15 = 24\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etotient function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e counts numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(15) = 8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(15) = 8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 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\u003eAn oblique hint is that I call solutions to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n-1) = phi(n+1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n-1) = \\\\phi(n+1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e “Beatriz numbers” in honor of a good friend and former co-worker of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/players/18927291\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eEmilyR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/players/8608872\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eJessicaR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \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":53810,"title":"List the Beatriz numbers","description":"Write a function to find integers  less than or equal to an input value  that solve the equation . The function  gives the sum of divisors of ; for example, . The totient function  counts numbers less than  that are relatively prime to . For example,  because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 1. \r\nAn oblique hint is that I call solutions to  “Beatriz numbers” in honor of a good friend and former co-worker of EmilyR and JessicaR. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.  ","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: 156px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407.5px 78px; transform-origin: 407.5px 78px; 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: 384.5px 42px; text-align: left; transform-origin: 384.5px 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: 98.275px 7.66667px; transform-origin: 98.275px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to find integers \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\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: 112.417px 7.66667px; transform-origin: 112.417px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e less than or equal to an input value \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003em\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: 74.2917px 7.66667px; transform-origin: 74.2917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that solve the equation \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n-1) = phi(n+1)\" style=\"width: 127px; height: 18.5px;\" width=\"127\" height=\"18.5\"\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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The function \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(x)\" style=\"width: 30.5px; height: 18.5px;\" width=\"30.5\" height=\"18.5\"\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: 31.8917px 7.66667px; transform-origin: 31.8917px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e gives the \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/46898\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003esum of divisors\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: 9.71667px 7.66667px; transform-origin: 9.71667px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e of \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 44.3333px 7.66667px; transform-origin: 44.3333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; for example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(15) = 1+3+5+15 = 24\" style=\"width: 178.5px; height: 18.5px;\" width=\"178.5\" height=\"18.5\"\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: 17.8833px 7.66667px; transform-origin: 17.8833px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/656\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration: underline; text-decoration-line: underline; \"\u003etotient function\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: 1.94167px 7.66667px; transform-origin: 1.94167px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAlCAYAAADyUO83AAACkUlEQVRoge2ZUbWDMAyGPw84wAAGpmAK5mAOcIAFNEwCHrAwDbPAfWhzyelaaEq594X/HB5GlzZJkz9tgAsXLly48G/ogAloKs13B14V59vF5J9ngWxt4wU3YD5h3i/cgcU/D6NsB7yBtrZSHg+cc0/FwOoAi7cb3A6VRI0FE9CfucCMM342yvW43T8bLU6/7szJF2xeboAP9pQpxYuTUuHB6oCbQa7HOeCvIDxVnWtefmKrMW8v+5dYcHx1GC1ut2UXF5xBvX+/l2udlykhP1k7RrZ7a0/YeeprcdnxF84Anf93VkIcEkqi5HJTpgNGnJNlvSEYl7EP6TCXalWEB+tui5d7pZAsqkkxFW4DtnxsWZ0phr79u9brNfh3W6ko+lq4CkgTnex2WMqmQMkQMl6C0Okz+ZVE7DA5QDwclrmG9E5Paqy2A25q7jmydo6s6VAkyoahtVX+Pmpsa85SyNzWymN2gPZ2yNgjcSM1B6QOHkcdIPJWRhd7squPMH6MsISMQiPDyhDDUQfos4fl7mGtPr+GhCTXkY4MXapSygmRlZzKtIMXXOnNhaybdSfQ4T9uKNElZLbyrIiNWR2fU2pjkLTNgjYmxfJhZOTeCsUA6+XpzbrjMR4Y2Y6qGeOFKBYBTeK9REWq9seU2bsLPP3T+f9rh4XngZHtaBCnm47fcmLTHZtY90eMH8knJZFJ/V9HoBy9Nbqd8RCSdub2WI8Lmw/OQH3IGfzvATuhST8gtSONWi+VKv3OuMbMN5cVIVX+SvAkP2WO4Ia9ZEahw65Wj+3sfp2QZ5XOU6r8HYE0Rs03tEyMVAp9SN8LjkK+C9RuWj6oaDykD0Y10FD3S86dyqkl3Z6J88L1woULF/4NP4rjEWlVkFcaAAAAAElFTkSuQmCC\" alt=\"phi(x)\" style=\"width: 32px; height: 18.5px;\" width=\"32\" height=\"18.5\"\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: 24.5px 7.66667px; transform-origin: 24.5px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e counts numbers less than \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 84.4px 7.66667px; transform-origin: 84.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e that are relatively prime to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ex\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: 46.6667px 7.66667px; transform-origin: 46.6667px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. For example, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIQAAAAlCAYAAACDBEHsAAAD80lEQVR4nO1bbbHDIBBcD3EQAzEQBVVQB3UQB7VQDZUQD7FQDbXQ94Ps5MoDwkHyQt6wM/xo2sBxt9wHUKCioqKioqKioqLi5OgAjACaowUB0MLI0h0tyJkxzu2W8G4KGVrl7zsAvac95v4kGgAv/C9SyDnviguAz9yuync7GMXbBvGhhTHgB/HGagC8hYx2G1dkK8FrpaKHmd8EYJjbE2bed+w0tzsW5WoGaGAEjfEqDRYisMUSYoCfDB+EV8wAP2FKxxWL4X3f7RKmp7nzSfneALMC13CZf3sRY8USgt4hJZQRr8z3j0ALo6M3/N6XnkLr1aMG/sAYLRY0lFYYudpjCHFDWCkxYB9nCh03LITwgbp8bjkwXc+a63UJExI29J6GEC8Yr9Ij3aANdlhJO0PqyTdv/uax5cB0O1rjvpDGTA0hJFnZpvm5lhxMzM4Cmej7DM7we8kdrIVZcVzlHxgDD/PzNUN18zspcVlDiBd+E4LtDb1H0ybNR0PO304sZVKZjBaLR3hiiVPMH2TSFypp+F5KPawNGf0s1x1ugsSuDq64FJm5gHKblowdvsvtaZaFOsxKlK+icxpCGodJm0wyXeUOsJSpKYmelhA25DzoKWIU3UOfOBNrZW9sS5mvTQp685wk25s40hvYpeMonruUze9TkEsIYNn/0FRHJHoKIa5YdnJzWqoRXaRI7q8VnUllMPN2eYJRfFciIYDvecXG0eyYewAamPD+wu+NvaRynMazq4hQuSnZGOozBVsRAlgUFFshnY0Q9ITyTKbHdy6l2l9h3HQlIJJtEjKH8CmvFEIwUdQQYtNNnJ1BPdu2s0Nm9D4EKwpXAkiW2Ua3K4+QoCnYkhAkfMz2OUvllBziiCqD8vrCAk9zVV5CZqWuwVzsk+5obYfsiCpDgh4iZoWQPLl7J39VZXDMkPeTi3e1bxkubIX5OpLvhFYS84+/2IcIgWEvZi8iR+YjqowYQkh7qQjhqyJszxF76plTwsUSooGZg0+JlCE2SeTeyVkgvbhPB/SQMSETgNtDNJ7n9BqxF0om7HuWISudwZKJSdW00odE6tnLkeDC9YVE5ojRYZCrQu5suW5HkQwPxCc+fEd7bU7mKCEPI/dCOIcBZk5vpaz0JtmHQH+MBt+kaMVz2lbtpXljiEqUir7Pn+/QJ4iaiyutGMtuj0AfV5hVIH87JMjKc5Czose3/p5I04MTvnIzBTeUf2eRxN39YuoZIROVlITQhXHDvvbAExtfIPlPUNWtkWCCV+IKvMLIVrIHOxS+c41c8H8ZJf0HoqQ/DhUL9d63AjyVK8EA/O9HCbIUC96GGlGme6+oqKioqKioqCgAPxSSChN/gD+jAAAAAElFTkSuQmCC\" alt=\"phi(15) = 8\" style=\"width: 66px; height: 18.5px;\" width=\"66\" height=\"18.5\"\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: 127.433px 7.66667px; transform-origin: 127.433px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 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: 384.5px 31.5px; text-align: left; transform-origin: 384.5px 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: 122.525px 7.66667px; transform-origin: 122.525px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAn oblique hint is that I call solutions to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" alt=\"sigma(n-1) = phi(n+1)\" style=\"width: 127px; height: 18.5px;\" width=\"127\" height=\"18.5\"\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: 182.817px 7.66667px; transform-origin: 182.817px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e “Beatriz numbers” in honor of a good friend and former co-worker of \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/players/18927291\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eEmilyR\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: 15.5583px 7.66667px; transform-origin: 15.5583px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/players/8608872\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eJessicaR\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: 268.4px 7.66667px; transform-origin: 268.4px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.\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 7.66667px; transform-origin: 3.88333px 7.66667px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e  \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function y = Beatriz(m)\r\n  y = f(m);\r\nend","test_suite":"%%\r\nm = 30;\r\ny_correct = [4 6 12 18 30];\r\nassert(isequal(Beatriz(m),y_correct))\r\n\r\n%% \r\nm = 200;\r\ny_correct = [4 6 12 18 30 42 60 72 102 108 138 150 180 192 198];\r\nassert(isequal(Beatriz(m),y_correct))\r\n\r\n%% \r\nm = 1000;\r\ny = Beatriz(m);\r\nhist_correct = [8 7 4 2 3 2 4 0 5 0];\r\ns_correct = 12118;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 10000;\r\ny = Beatriz(m);\r\nhist_correct = [35 26 20 22 23 17 19 13 14 16];\r\ns_correct = 863206;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 100000;\r\ny = Beatriz(m);\r\nhist_correct = [205 137 125 124 114 106 94 102 109 108];\r\ns_correct = 54557284;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 1e6;\r\ny = Beatriz(m);\r\nhist_correct = [1224 936 834 810 761 766 730 705 706 697];\r\ns_correct = 3712183324;\r\nhist1 = histcounts(y,0:m/10:m);\r\nassert(isequal(hist1,hist_correct) \u0026\u0026 isequal(sum(y),s_correct))\r\n\r\n%% \r\nm = 1e7;\r\ny = Beatriz(m);\r\na = y(1:2949:end);\r\na_correct = [4 294180 675840 1093638 1527678 1980660 2458368 2954772 3442122 3941688 4461642 4994790 5535240 6076308 6624972 7185888 7734828 8288340 8852070 9423180];\r\nassert(isequal(a,a_correct))\r\n\r\n%%\r\ny = Beatriz(1e9);\r\ns = sum(1./(y-1)+1./(y+1));\r\nB2 = 1.902160583104;\r\ner_correct = 0.127424625465465;\r\nassert(B2-s-er_correct\u003c1e-9)\r\n\r\n%%\r\nfiletext = fileread('Beatriz.m');\r\nillegal = contains(filetext, 'assignin') || contains(filetext, 'regexp') || contains(filetext, 'oeis'); \r\nassert(~illegal)\r\n","published":true,"deleted":false,"likes_count":3,"comments_count":0,"created_by":46909,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":9,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2022-01-10T05:12:56.000Z","updated_at":"2025-12-07T06:51:03.000Z","published_at":"2022-01-10T05:18:11.000Z","restored_at":null,"restored_by":null,"spam":false,"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\u003eWrite a function to find integers \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"n\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e less than or equal to an input value \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"m\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003em\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that solve the equation \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n-1) = phi(n+1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n-1) = \\\\phi(n+1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The function \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e gives the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/46898\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esum of divisors\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; for example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(15) = 1+3+5+15 = 24\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(15) = 1+3+5+15 = 24\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/656\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003etotient function\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(x)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(x)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e counts numbers less than \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e that are relatively prime to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"x\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. For example, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"phi(15) = 8\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\phi(15) = 8\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e because 1, 2, 4, 7, 8, 11, 13, and 14 are relatively prime to 15; that is, the greatest common divisor of 15 and each of the eight numbers is 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\u003eAn oblique hint is that I call solutions to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"sigma(n-1) = phi(n+1)\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e\\\\sigma(n-1) = \\\\phi(n+1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e “Beatriz numbers” in honor of a good friend and former co-worker of \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/players/18927291\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eEmilyR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/players/8608872\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eJessicaR\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e. The natural search will not help, unless you are interested in combinatorics related to Higham’s conjecture.\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e  \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\"}]}"}],"term":"tag:\"misdirection\"","current_player_id":null,"fields":[{"name":"page","type":"integer","callback":null,"default":1,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"per_page","type":"integer","callback":null,"default":50,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"sort","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":null,"prepend":true},{"name":"body","type":"text","callback":null,"default":"*:*","directive":null,"facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":false},{"name":"group","type":"string","callback":null,"default":null,"directive":"group","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"difficulty_rating_bin","type":"string","callback":null,"default":null,"directive":"difficulty_rating_bin","facet":true,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"id","type":"integer","callback":null,"default":null,"directive":"id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"tag","type":"string","callback":null,"default":null,"directive":"tag","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"product","type":"string","callback":null,"default":null,"directive":"product","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_at","type":"timeframe","callback":{},"default":null,"directive":"created_at","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"profile_id","type":"integer","callback":null,"default":null,"directive":"author_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"created_by","type":"string","callback":null,"default":null,"directive":"author","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player_id","type":"integer","callback":null,"default":null,"directive":"solver_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"player","type":"string","callback":null,"default":null,"directive":"solver","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"solvers_count","type":"integer","callback":null,"default":null,"directive":"solvers_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"comments_count","type":"integer","callback":null,"default":null,"directive":"comments_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"likes_count","type":"integer","callback":null,"default":null,"directive":"likes_count","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leader_id","type":"integer","callback":null,"default":null,"directive":"leader_id","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true},{"name":"leading_solution","type":"integer","callback":null,"default":null,"directive":"leading_solution","facet":null,"facet_method":"and","operator":null,"param":"term","static":null,"prepend":true}],"filters":[{"name":"asset_type","type":"string","callback":null,"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":null,"static":"\"cody:problem\"","prepend":true},{"name":"profile_id","type":"integer","callback":{},"default":null,"directive":null,"facet":null,"facet_method":"and","operator":null,"param":"author_id","static":null,"prepend":true}],"query":{"params":{"per_page":50,"term":"tag:\"misdirection\"","current_player":null,"sort":"map(difficulty_value,0,0,999) 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