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A379616 Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

%I A379616 #7 Dec 28 2024 09:11:30
%S A379616 1,3,12,60,20,30,120,40,40,360,360,72,504,126,504,1512,1512,7560,1512,
%T A379616 7560,30240,30240,30240,30240,393120,393120,393120,393120,393120,
%U A379616 393120,196560,28080,14040,4680,9360,46800,889200,889200,6224400,6224400,889200,1778400
%N A379616 Denominators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).
%H A379616 Amiram Eldar, <a href="/A379616/b379616.txt">Table of n, a(n) for n = 1..1000</a>
%H A379616 V. Sitaramaiah and M. V. Subbarao, <a href="https://informaticsjournals.co.in/index.php/jims/article/view/21936">Asymptotic formulae for sums of reciprocals of some multiplicative functions</a>, J. Indian Math. Soc., Vol. 57 (1991), pp. 153-167.
%H A379616 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 A379616 a(n) = denominator(Sum_{k=1..n} 1/A188999(k)).
%t A379616 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/bsigma[n], {n, 1, 50}]]]
%o A379616 (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 A379616 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / bsigma(k); print1(denominator(s), ", "))};
%Y A379616 Cf. A188999, A307159, A370904, A379615 (numerators), A379618.
%K A379616 nonn,easy,frac
%O A379616 1,2
%A A379616 _Amiram Eldar_, Dec 27 2024