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

Author

Emeric Deutsch, Mar 12 2005

Keywords

• frac
• nonn

Examples

			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.
		

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.

Crossrefs

Cf. A000005, A104529 (denominators).

Programs

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 *)

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