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

%I A379361 #7 Dec 22 2024 16:52:47
%S A379361 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,
%T A379361 52,89,119,89,119,803,1013,803,1013,1921,2341,1921,2341,2201,2621,
%U A379361 2201,2621,2411,2621,2201,2621,2537,2747,2537,2957,2747,3167,1009,1149,3307
%N A379361 Numerators of the partial alternating sums of the reciprocals of the number of abelian groups function (A000688).
%H A379361 Amiram Eldar, <a href="/A379361/b379361.txt">Table of n, a(n) for n = 1..10000</a>
%H A379361 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.8, pp. 27-28.
%F A379361 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A000688(k)).
%F A379361 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).
%e A379361 Fractions begin with 1, 0, 1, 1/2, 3/2, 1/2, 3/2, 7/6, 5/3, 2/3, 5/3, 7/6, ...
%t A379361 Numerator[Accumulate[Table[(-1)^(n+1)/FiniteAbelianGroupCount[n], {n, 1, 100}]]]
%o A379361 (PARI) f(n) = vecprod(apply(numbpart, factor(n)[, 2]));
%o A379361 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / f(k); print1(numerator(s), ", "))};
%Y A379361 Cf. A000041, A000688, A063966, A084911, A370897, A379359, A379362 (denominators).
%K A379361 nonn,easy,frac
%O A379361 1,5
%A A379361 _Amiram Eldar_, Dec 21 2024