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A379621 Numerators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

%I A379621 #9 Dec 28 2024 09:08:59
%S A379621 1,0,1,1,5,-1,1,-7,11,-47,-13,-61,-29,-157,-209,-3139,-5123,-1109,
%T A379621 -2887,-3547,-2887,-3679,-3319,-4111,-26137,-30757,-5597,-2071,-277,
%U A379621 -343,-1627,-12269,-2269,-625,-391,-1261,-3629,-3937,-1853,-4979,-19223,-21533,-20873,-21797
%N A379621 Numerators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).
%H A379621 Amiram Eldar, <a href="/A379621/b379621.txt">Table of n, a(n) for n = 1..1000</a>
%H A379621 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 4.14, p. 35.
%F A379621 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A206369(k)).
%F A379621 a(n)/A379622(n) = A * log(n) + B + O(1/n^u), where u > 0, and A and B are constants.
%e A379621 Fractions begin with 1, 0, 1/2, 1/6, 5/12, -1/12, 1/12, -7/60, 11/420, -47/210, -13/105, -61/210, ...
%t A379621 f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/beta[n], {n, 1, 50}]]]
%o A379621 (PARI) beta(n) = {my(f = factor(n)); prod(i=1, #f~, p = f[i, 1]; e = f[i, 2]; sum(k = 0, e, (-1)^(e-k)*p^k)); }
%o A379621 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / beta(k); print1(numerator(s), ", "))};
%Y A379621 Cf. A206369, A370905, A370906, A379619, A379622 (denominators).
%K A379621 sign,easy,frac
%O A379621 1,5
%A A379621 _Amiram Eldar_, Dec 27 2024