A104528 Numerator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.

1, 3, 2, 7, 17, 37, 43, 23, 25, 53, 59, 61, 67, 35, 73, 377, 407, 139, 149, 457, 118, 487, 517, 1049, 363, 373, 383, 1169, 1229, 311, 163, 331, 677, 173, 707, 2141, 2231, 569, 2321, 4687, 4867, 614, 1273, 644, 1303, 2651, 2741, 2759, 2819, 2849, 1447, 731

Offset

1,2

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Srinivasan Ramanujan, Some formulae in the analytic theory of numbers, Messenger of Math., Vol. 45 (1916), pp. 81-84.

B. M. Wilson, Proofs of some formulae enunciated by Ramanujan, Proceedings of the London Mathematical Society, Volume s2-21, Issue 1 (1923), pp. 235-255.

Formula

Sum_{k=1..n} a(k)/A104529(k) ~ (n/sqrt(log(n)) * (c_0 + c_1/log(n) + .... + c_k/log(n)^k + O(1/log(n)^(k+1))), where c_0 = (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Ramanujan, 1916; Wilson, 1923). - Amiram Eldar, Oct 14 2022

Example

Fractions begin with 1, 3/2, 2, 7/3, 17/6, 37/12, 43/12, 23/6, 25/6, 53/12, 59/12, 61/12, ...
a(4) = 7 because 1/tau(1) + 1/tau(2) + 1/tau(3) + 1/tau(4) = 1/1 + 1/2 + 1/2 + 1/3 = 7/3.

Maple

with(numtheory): a:=n->numer(sum(1/tau(k), k=1..n)): seq(a(n), n=1..57);

Mathematica

Numerator[Accumulate[1/Array[DivisorSigma[0, #] &, 50]]] (* Amiram Eldar, Oct 14 2022 *)

Crossrefs

Cf. A000005, A104529 (denominators).

Sequence in context: A363399 A049970 A344211 * A177115 A316087 A196537

Adjacent sequences: A104525 A104526 A104527 * A104529 A104530 A104531

Keyword

frac,nonn

Author

Emeric Deutsch, Mar 12 2005

Status

approved