A212717 Numerator of Sum_{k=1..n} 1/sigma(k).

1, 4, 19, 145, 53, 83, 353, 607, 8171, 75359, 78089, 79259, 11657, 2963, 12047, 378137, 386197, 389917, 397171, 2804377, 11344453, 11457293, 11626553, 11694257, 11825297, 11922017, 12023573, 12096113, 12231521, 12287941, 6207443, 6239683, 3140999, 9479417

Offset

1,2

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Formula

Sum_{k=1..n} a(k)/A212718(k) ~ c * (log(n) + gamma + Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p))) + O(log(n)^(2/3)*log(log(n))^(4/3)/n), where alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), and c = Product_{p prime} alpha(p) (Sita Ramaiah and Suryanarayana, 1979). - Amiram Eldar, Oct 16 2022

Example

1, 4/3, 19/12, 145/84, 53/28, 83/42, 353/168, ...

Maple

with(numtheory): a:=n->numer(sum(1/sigma(k), k=1..n)): seq(a(n), n=1..50);

Mathematica

Numerator[Table[Sum[1/DivisorSigma[1, k], {k, 1, n}], {n, 1, 50}]]
Accumulate[1/DivisorSigma[1, Range[40]]]//Numerator (* Harvey P. Dale, Aug 13 2023 *)

Crossrefs

Cf. A000203, A212718 (denominators).

Sequence in context: A305725 A208992 A321388 * A067577 A225904 A203010

Adjacent sequences: A212714 A212715 A212716 * A212718 A212719 A212720

Keyword

nonn,frac

Author

Michel Lagneau, May 25 2012

Status

approved