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A379361
Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).
3
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
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
Sequence in context: A107461 A035619 A280995 * A092689 A281553 A064434
KEYWORD
nonn,easy,frac
AUTHOR
Amiram Eldar, Dec 21 2024
STATUS
approved