{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:19:47Z","timestamp":1776788387739,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"264","license":[{"start":{"date-parts":[[2009,5,20]],"date-time":"2009-05-20T00:00:00Z","timestamp":1242777600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This note contains a new algorithm for computing a function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    introduced by Erd\u0151s to measure the minimal gap size in the sequence of integers at least one of whose prime factors exceeds\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                    . This algorithm enables us to show that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is not monotone, verifying a conjecture of Ecklund and Eggleton.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02134-0","type":"journal-article","created":{"date-parts":[[2008,7,25]],"date-time":"2008-07-25T12:55:42Z","timestamp":1216990542000},"page":"2455-2459","source":"Crossref","is-referenced-by-count":5,"title":["Prime factors of consecutive integers"],"prefix":"10.1090","volume":"77","author":[{"given":"Mark","family":"Bauer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael","family":"Bennett","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,5,20]]},"reference":[{"issue":"1-4","key":"1","doi-asserted-by":"publisher","first-page":"30","DOI":"10.2307\/1967797","article-title":"On the numerical factors of the arithmetic forms \ud835\udefc\u207f\u00b1\ud835\udefd\u207f","volume":"15","author":"Carmichael, R. D.","year":"1913","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"1082","DOI":"10.2307\/2317422","article-title":"Prime factors of consecutive integers","volume":"79","author":"Ecklund, E. F., Jr.","year":"1972","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"3","volume-title":"Proceedings of the Manitoba Conference on Numerical Mathematics","year":"1971"},{"key":"4","first-page":"161","article-title":"Consecutive integers all of whose prime factors belong to a given set","author":"Ecklund, E. F., Jr.","year":"1974"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/s1446788700013318","article-title":"Consecutive integers with no large prime factors","volume":"22","author":"Eggleton, R. B.","year":"1976","journal-title":"J. Austral. Math. Soc. Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0263-6115","issn-type":"print"},{"key":"6","unstructured":"P. Erd\u0151s. On consecutive integers. Nieuw Arch. Wisk. 3 (1955), 124\u2013128."},{"key":"7","series-title":"Monographies de L'Enseignement Math\\'{e}matique [Monographs of L'Enseignement Math\\'{e}matique]","volume-title":"Old and new problems and results in combinatorial number theory","volume":"28","author":"Erd\u0151s, P.","year":"1980"},{"key":"8","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-26677-0","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387208607","edition":"3"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"419","DOI":"10.2307\/1968235","article-title":"An extended theory of Lucas\u2019 functions","volume":"31","author":"Lehmer, D. H.","year":"1930","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1215\/ijm\/1256067456","article-title":"On a problem of St\u00f6rmer","volume":"8","author":"Lehmer, D. H.","year":"1964","journal-title":"Illinois J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0019-2082","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"19","DOI":"10.2307\/2313306","article-title":"The prime factors of consecutive integers","volume":"72","author":"Lehmer, D. H.","year":"1965","journal-title":"Amer. Math. Monthly","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9890","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","unstructured":"R. Rankin. The difference between consecutive prime numbers. J. London Math. Soc. 13 (1938), 242\u2013247.","DOI":"10.1112\/jlms\/s1-13.4.242"},{"key":"13","unstructured":"J. Schur. Einege S\u00e4tze \u00fcber Primzahlen mit Anwendung auf Irreduzibilit\u00e4tsfragen. S.-B. Deutsch. Akad. Wiss Berlin Kl. Math. Phys. Tech. 23 (1929), 1\u201324."},{"key":"14","doi-asserted-by":"publisher","first-page":"365","DOI":"10.4064\/aa-25-4-365-373","article-title":"On gaps between numbers with a large prime factor. II","volume":"25","author":"Shorey, T. N.","year":"1973","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"15","unstructured":"J. J. Sylvester. On arithmetical series. Messenger Math. 21 (1892), 1\u201319, 87\u2013120."},{"key":"16","doi-asserted-by":"publisher","first-page":"896","DOI":"10.2307\/2311692","article-title":"A conjecture of Erd\u0151s concerning consecutive integers","volume":"68","author":"Utz, W. R.","year":"1961","journal-title":"Amer. Math. 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