cp's OEIS Frontend

A379622 Denominators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).

%I A379622 #8 Dec 28 2024 09:09:19
%S A379622 1,1,2,6,12,12,12,60,420,210,105,210,140,420,840,9240,18480,2640,7920,
%T A379622 7920,7920,7920,7920,7920,55440,55440,11088,3696,528,528,2640,18480,
%U A379622 3696,924,616,1848,5544,5544,2772,6930,27720,27720,27720,27720,27720,27720,637560
%N A379622 Denominators of the partial alternating sums of the reciprocals of the alternating sum of divisors function (A206369).
%H A379622 Amiram Eldar, <a href="/A379622/b379622.txt">Table of n, a(n) for n = 1..1000</a>
%H A379622 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 A379622 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A206369(k)).
%t A379622 f[p_, e_] := Sum[(-1)^(e-k)*p^k, {k, 0, e}]; beta[1] = 1; beta[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/beta[n], {n, 1, 50}]]]
%o A379622 (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 A379622 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / beta(k); print1(denominator(s), ", "))};
%Y A379622 Cf. A206369, A370905, A370906, A379620, A379621 (numerators).
%K A379622 nonn,easy,frac
%O A379622 1,3
%A A379622 _Amiram Eldar_, Dec 27 2024