Banks 2005 nonaliquots robbins numbers

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Banks, William D. and Luca, Florian, Nonaliquots and {R}obbins numbers. Colloq. Math. 103 (2005), 27--32. The file's text layer carries no copyright or license line; the publisher's issue listing marks the article "Free download under CC-BY license", as it marks every article in the issue, and names no Creative Commons version or URL (https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/103/1, read 2026-10-02; the article's own page was not opened), so the term is the Creative Commons Attribution license with its version unstated; the site footer "Copyright © 2026 by IMPAN. All rights reserved." speaks for the site, not the article.

An integer m is nonaliquot if m = sigma(n) - n has no solution; Erdos proved the nonaliquot numbers have positive lower density but gave no numerical value. Theorem 1 supplies one: the counting function N_a(x) of nonaliquot numbers up to x satisfies #N_a(x) >= (x/48)(1 + o(1)), the proof taking the multiples of 12 up to x and showing that at most (x/16)(1 + o(1)) of them can be written as sigma(n) - n, so that at least a quarter of them are nonaliquot, by sorting the possible shapes of n. Theorem 2 treats Robbins numbers, the integers m not of the form (p-1)/2 - phi(p-1) for an odd prime p, and shows #N_r(x) >= (x/3)(1 + o(1)), so they too have positive lower density; this strengthens Luca and Walsh's result that infinitely many exist. The method throughout is elementary counting with the Euler and divisor-sum functions plus standard prime-counting estimates in arithmetic progressions. Problem 418 asks about the integers not of the form n - phi(n); this paper treats the companion function sigma(n) - n, so its 1/48 is adjacent to that question and does not answer it.

Source: https://www.impan.pl/en/publishing-house/journals-and-series/colloquium-mathematicum/all/103/1.

Bears on. #418

Results to transcribe.

• Theorem 1: The number of nonaliquot m <= x (those with no n satisfying sigma(n) - n = m) is at least (x/48)(1 + o(1)), an explicit form of Erdos's positive-density theorem.
• Theorem 2: The set of Robbins numbers, integers m never equal to (p-1)/2 - phi(p-1) for an odd prime p, satisfies #N_r(x) >= (x/3)(1 + o(1)), hence has positive lower density.