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A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).

%I A379617 #7 Dec 28 2024 09:10:55
%S A379617 1,2,11,43,53,4,37,103,23,65,71,337,2539,1217,2539,7337,7757,1501,
%T A379617 7883,7631,31469,30629,31889,6277,84625,82753,423593,82753,426869,
%U A379617 421409,216847,213727,108911,11899,24253,119081,2317139,760853,773203,6889667,7037867,13946059
%N A379617 Numerators of the partial alternating sums of the reciprocals of the sum of bi-unitary divisors function (A188999).
%H A379617 Amiram Eldar, <a href="/A379617/b379617.txt">Table of n, a(n) for n = 1..1000</a>
%H A379617 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 A379617 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A188999(k)).
%F A379617 a(n)/A379618(n) = A * log(n) + B + O(log(n)^(14/3) * log(log(n))^(4/3) * n^c), where c = log(9/10)/log(2) = -0.152003..., and A and B are constants.
%e A379617 Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 103/120, 23/24, 65/72, 71/72, 337/360, ...
%t A379617 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)^(n+1)/bsigma[n], {n, 1, 50}]]]
%o A379617 (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 A379617 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / bsigma(k); print1(numerator(s), ", "))};
%Y A379617 Cf. A188999, A307159, A370904, A379615, A379618 (denominators).
%K A379617 nonn,easy,frac
%O A379617 1,2
%A A379617 _Amiram Eldar_, Dec 27 2024