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

1,5

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.

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).

Example

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

Mathematica

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

Prog

(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), ", "))};

Crossrefs

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

Sequence in context: A107461 A035619 A280995 * A092689 A281553 A064434

Adjacent sequences: A379358 A379359 A379360 * A379362 A379363 A379364

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 21 2024

Status

approved