A330899 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.

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 22, 24, 28, 30, 36, 42, 48, 56, 60, 72, 84, 90, 96, 100, 108, 112, 120, 144, 156, 168, 180, 192, 210, 240, 276, 280, 288, 300, 312, 324, 330, 336, 360, 396, 408, 420, 480, 528, 540, 576, 600, 630, 660, 672, 720, 756, 792

Offset: 1

Author

Amiram Eldar, May 01 2020

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000203, A013661, A284648, A284650.

Programs

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