{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:37:38Z","timestamp":1776724658974,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"221","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi (n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    denote Euler\u2019s totient function, i.e., the number of positive integers\u00a0\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"greater-than n\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">&gt;n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and prime to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We study pairs of positive integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis a 0 comma a 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(a_{0},a_{1})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a 0 less-than-or-equal-to a 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">a_{0}\\le a_{1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi left-parenthesis a 0 right-parenthesis equals phi left-parenthesis a 1 right-parenthesis equals left-parenthesis a 0 plus a 1 right-parenthesis slash k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>a<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi (a_{0})=\\phi (a_{1})=(a_{0}+a_{1})\/k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for some integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k\\ge 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We call these numbers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013\n                    <italic>amicable pairs with multiplier<\/italic>\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , analogously to Carmichael\u2019s multiply amicable pairs for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"sigma\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c3\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\sigma<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013function (which sums all the divisors of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). We have computed all the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs with larger member\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"less-than-or-equal-to 10 Superscript 9\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>9<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\le 10^{9}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and found\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"812\">\n                        <mml:semantics>\n                          <mml:mn>812<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">812<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs can be associated. Among these\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"812\">\n                        <mml:semantics>\n                          <mml:mn>812<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">812<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    pairs there are\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"499\">\n                        <mml:semantics>\n                          <mml:mn>499<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">499<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    so-called primitive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs. We present a table of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"58\">\n                        <mml:semantics>\n                          <mml:mn>58<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">58<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    primitive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs for which the larger member does not exceed\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"10 Superscript 6\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mn>10<\/mml:mn>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>6<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">10^{6}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Next,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs with a given prime structure are studied. It is proved that a relatively prime\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pair has at least twelve distinct prime factors and that, with the exception of the pair\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 4 comma 6 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>6<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(4,6)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , if one member of a\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    \u2013amicable pairs with larger member\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"greater-than 10 Superscript 9\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>10<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>9<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">&gt;10^{9}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the largest pair consisting of two 46-digit numbers.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-00933-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"399-411","source":"Crossref","is-referenced-by-count":3,"title":["On \ud835\udf19-amicable pairs"],"prefix":"10.1090","volume":"67","author":[{"given":"Graeme","family":"Cohen","sequence":"first","affiliation":[]},{"given":"Herman","family":"te Riele","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1002\/mana.3210630125","article-title":"Eine Schranke f\u00fcr befreundete Zahlen mit gegebener Teileranzahl","volume":"63","author":"Borho, Walter","year":"1974","journal-title":"Math. Nachr.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-584X","issn-type":"print"},{"issue":"153","key":"2","doi-asserted-by":"publisher","first-page":"303","DOI":"10.2307\/2007752","article-title":"Some large primes and amicable numbers","volume":"36","author":"Borho, W.","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"3","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1016\/0315-0860(89)90084-0","article-title":"Notes on Th\u0101bit ibn Qurra and his rule for amicable numbers","volume":"16","author":"Brentjes, Sonja","year":"1989","journal-title":"Historia Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0315-0860","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"R. D. Carmichael, Review of History of the Theory of Numbers, Amer. Math. Monthly 26 (1919), 396\u2013403.","DOI":"10.2307\/2971917"},{"key":"5","unstructured":"G. L. Cohen and H. J. J. te Riele, On \ud835\udf19\u2013amicable pairs (\u030awith appendix)\u030a, Research Report R95-9 (December 1995), School of Mathematical Sciences, University of Technology, Sydney, and CWI-Report NM-R9524 (November 1995), CWI Amsterdam, \\url{ftp:\/\/ftp.cwi.nl\/pub\/CWIreports\/NW\/NM-R9524.ps.Z}."},{"key":"6","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3585-4","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387942890","edition":"2"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"114","DOI":"10.4153\/CMB-1982-015-x","article-title":"The solution of a special arithmetic equation","volume":"25","author":"Hausman, Miriam","year":"1982","journal-title":"Canad. Math. Bull.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-4395","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"T. E. Mason, On amicable numbers and their generalizations, Amer. Math. Monthly 28 (1921), 195\u2013200.","DOI":"10.1080\/00029890.1921.11986031"},{"key":"9","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0099436","volume-title":"Number theory, Noordwijkerhout 1983","volume":"1068","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/3540133569"},{"issue":"175","key":"10","doi-asserted-by":"publisher","first-page":"361","DOI":"10.2307\/2008101","article-title":"Computation of all the amicable pairs below 10\u00b9\u2070","volume":"47","author":"te Riele, H. J. J.","year":"1986","journal-title":"Math. 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