A379362 Denominators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).

1, 1, 1, 2, 2, 2, 2, 6, 3, 3, 3, 6, 6, 6, 6, 30, 30, 15, 15, 30, 30, 30, 30, 30, 15, 15, 15, 30, 30, 30, 30, 210, 210, 210, 210, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 420, 140, 140, 420, 420, 420, 420, 420, 420, 420

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• nonn
• easy
• frac

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. A000688, A063966, A370897, A379360, A379361 (numerators).

Programs

Mathematica

Denominator[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(denominator(s), ", "))};

Formula

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