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A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

%I A379615 #8 Dec 28 2024 09:11:16
%S A379615 1,4,19,107,39,61,259,89,93,857,887,181,1303,331,1345,4091,4175,21127,
%T A379615 4301,21757,87973,88813,90073,90577,1192621,1201981,1211809,1221637,
%U A379615 1234741,1240201,626243,89909,45247,15169,30533,153601,2941819,2956639,20807623,20876783
%N A379615 Numerators of the partial sums of the reciprocals of the sum of bi-unitary divisors function (A188999).
%H A379615 Amiram Eldar, <a href="/A379615/b379615.txt">Table of n, a(n) for n = 1..1000</a>
%H A379615 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 A379615 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 A379615 a(n) = numerator(Sum_{k=1..n} 1/A188999(k)).
%F A379615 a(n)/A379616(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) / n), where A and B are constants.
%e A379615 Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 89/40, 93/40, 857/360, 887/360, 181/72, ...
%t A379615 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]; Numerator[Accumulate[Table[1/bsigma[n], {n, 1, 50}]]]
%o A379615 (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 A379615 list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / bsigma(k); print1(numerator(s), ", "))};
%Y A379615 Cf. A188999, A307159, A370904, A379616 (denominators), A379617.
%K A379615 nonn,easy,frac
%O A379615 1,2
%A A379615 _Amiram Eldar_, Dec 27 2024