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%I #12 May 01 2020 09:58:53
%S 1,2,3,4,6,8,10,12,16,18,20,22,24,28,30,36,42,48,56,60,72,84,90,96,
%T 100,108,112,120,144,156,168,180,192,210,240,276,280,288,300,312,324,
%U 330,336,360,396,408,420,480,528,540,576,600,630,660,672,720,756,792
%N Numbers m such that (1/m) * Sum_{k=1..m} sigma(k)/k sets a record value, where sigma(k) is the sum of divisors of k.
%C Numbers m such that the mean of the abundancy index sigma(k)/k in the range 1..m is closer to the asymptotic mean Pi^2/6 than the mean in any smaller range.
%C Since (1/m) * Sum_{k=1..m} sigma(k)/k < Pi^2/6 for all m, and the limit is Pi^2/6 as m -> infinity, this sequence is infinite.
%H Amiram Eldar, <a href="/A330899/b330899.txt">Table of n, a(n) for n = 1..600</a>
%H R. A. MacLeod, <a href="https://doi.org/10.1017/S0004972700027076">Extreme values for divisor functions</a>, Bulletin of the Australian Mathematical Society, Vol. 37, No. 3, (1988), pp. 447-465. See Theorem 9 (iii), p. 463.
%H Y. -F. S. Pétermann, <a href="https://doi.org/10.1007/BF01630684">An Omega-theorem for an error term related to the sum-of-divisors function</a>, Monatshefte für Mathematik, Vol. 103, No. 2 (1987), pp. 145-157.
%e The mean abundancy in the range 1..m for m = 1, 2, ..., 6 is 1, 1.25, 1.277..., 1.395..., 1.356..., 1.463..., so the record values occur at 1, 2, 3, 4 and 6.
%t seq = {}; s = 0; rm = 0; Do[s += DivisorSigma[1, n]/n; r = s/n; If[r > rm, rm = r; AppendTo[seq, n]], {n, 1, 1000}]; seq
%Y Cf. A000203, A013661, A284648, A284650.
%K nonn
%O 1,2
%A _Amiram Eldar_, May 01 2020
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