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A104528
Numerator of Sum_{k=1..n} 1/tau(k), where tau(k) is the number of divisors function.
5
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
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
KEYWORD
frac,nonn
AUTHOR
Emeric Deutsch, Mar 12 2005
STATUS
approved