OFFSET
1,2
COMMENTS
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.
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..600
R. A. MacLeod, Extreme values for divisor functions, Bulletin of the Australian Mathematical Society, Vol. 37, No. 3, (1988), pp. 447-465. See Theorem 9 (iii), p. 463.
Y. -F. S. Pétermann, An Omega-theorem for an error term related to the sum-of-divisors function, Monatshefte für Mathematik, Vol. 103, No. 2 (1987), pp. 145-157.
EXAMPLE
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.
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 01 2020
STATUS
approved