cp's OEIS Frontend

A379618 Denominators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

%I A379618 #7 Dec 28 2024 09:10:39
%S A379618 1,3,12,60,60,5,40,120,24,72,72,360,2520,1260,2520,7560,7560,1512,
%T A379618 7560,7560,30240,30240,30240,6048,78624,78624,393120,78624,393120,
%U A379618 393120,196560,196560,98280,10920,21840,109200,2074800,691600,691600,6224400,6224400,12448800
%N A379618 Denominators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).
%H A379618 Amiram Eldar, <a href="/A379618/b379618.txt">Table of n, a(n) for n = 1..1000</a>
%H A379618 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 4.13, p. 34.
%F A379618 a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).
%t A379618 f[p_, e_] := (p^(e+1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; Denominator[Accumulate[Table[(-1)^(n+1)/bsigma[n], {n, 1, 50}]]]
%o A379618 (PARI) bsigma(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1) - 1)/(f[i, 1] - 1) - if(!(f[i, 2] % 2), f[i, 1]^(f[i, 2]/2)));}
%o A379618 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(denominator(s), ", "))};
%Y A379618 Cf. A188999, A307159, A370904, A379616, A379617 (numerators).
%K A379618 nonn,easy,frac
%O A379618 1,2
%A A379618 _Amiram Eldar_, Dec 27 2024