Multiperfect numbers on lines of the Pascal triangle

  1. Luca, F. 2
  2. Varona, J.L. 1
  1. 1 Universidad de La Rioja
    info

    Universidad de La Rioja

    Logroño, España

    ROR https://ror.org/0553yr311

  2. 2 Universidad Nacional Autónoma de México
    info

    Universidad Nacional Autónoma de México

    Ciudad de México, México

    ROR https://ror.org/01tmp8f25

Journal:
Journal of Number Theory

ISSN: 0022-314X

Year of publication: 2009

Volume: 129

Issue: 5

Pages: 1136-1148

Type: Article

DOI: 10.1016/J.JNT.2008.10.003 SCOPUS: 2-s2.0-62349107107 WoS: WOS:000265753700013 GOOGLE SCHOLAR lock_openOpen access editor

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Abstract

A number n is said to be multiperfect (or multiply perfect) if n divides its sum of divisors σ (n). In this paper, we study the multiperfect numbers on straight lines through the Pascal triangle. Except for the lines parallel to the edges, we show that all other lines through the Pascal triangle contain at most finitely many multiperfect numbers. We also study the distribution of the numbers σ (n) / n whenever the positive integer n ranges through the binomial coefficients on a fixed line through the Pascal triangle.