%I #10 Jan 08 2025 10:13:08
%S 1,3,2,7,17,37,43,23,25,53,59,61,67,35,73,377,407,139,149,457,118,487,
%T 517,1049,363,373,383,1169,1229,311,163,331,677,173,707,2141,2231,569,
%U 2321,4687,4867,614,1273,644,1303,2651,2741,2759,2819,2849,1447,731
%N Numerator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.
%H Amiram Eldar, <a href="/A104528/b104528.txt">Table of n, a(n) for n = 1..10000</a>
%H Srinivasan Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram17.html">Some formulae in the analytic theory of numbers</a>, Messenger of Math., Vol. 45 (1916), pp. 81-84.
%H B. M. Wilson, <a href="https://doi.org/10.1112/plms/s2-21.1.235">Proofs of some formulae enunciated by Ramanujan</a>, Proceedings of the London Mathematical Society, Volume s2-21, Issue 1 (1923), pp. 235-255.
%F 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
%e 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, ...
%e 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.
%p with(numtheory): a:=n->numer(sum(1/tau(k),k=1..n)): seq(a(n),n=1..57);
%t Numerator[Accumulate[1/Array[DivisorSigma[0, #] &, 50]]] (* _Amiram Eldar_, Oct 14 2022 *)
%Y Cf. A000005, A104529 (denominators).
%K frac,nonn
%O 1,2
%A _Emeric Deutsch_, Mar 12 2005