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

1,2

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

Cf. A000203, A013661, A284648, A284650.

Sequence in context: A302833 A093717 A359754 * A325796 A279029 A316886

Adjacent sequences: A330896 A330897 A330898 * A330900 A330901 A330902

Keyword

nonn

Author

Amiram Eldar, May 01 2020

Status

approved