A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

1, 0, 1, 1, 3, 1, 3, 7, 5, 2, 5, 7, 13, 7, 13, 59, 89, 37, 52, 89, 119, 89, 119, 109, 62, 47, 52, 89, 119, 89, 119, 803, 1013, 803, 1013, 1921, 2341, 1921, 2341, 2201, 2621, 2201, 2621, 2411, 2621, 2201, 2621, 2537, 2747, 2537, 2957, 2747, 3167, 1009, 1149, 3307

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.

Crossrefs

Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).

Programs

Mathematica

Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]

PARI

f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).

a(n)/A379362(n) ~ D * c * n, where D = A084911, c = 2/(1 + Sum_{k>=1} 1/(P(k)*2^k)) - 1 = 0.18634377034863729099..., and P(k) = A000041(k).