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A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

%I A379515 #8 Dec 24 2024 07:28:24
%S A379515 1,2,11,43,53,4,37,293,329,103,113,107,809,129,809,12913,41119,39691,
%T A379515 41833,8081,33395,32443,33871,10973,148361,48275,7149,34861,108119,
%U A379515 319937,164941,1761311,112361,662011,5405483,26502319,516671461,508357441,3620857237,3556192637
%N A379515 Numerators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).
%H A379515 Amiram Eldar, <a href="/A379515/b379515.txt">Table of n, a(n) for n = 1..1000</a>
%H A379515 Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Bordelles/bord14.html">An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
%H A379515 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 4.9, pp. 28-29.
%F A379515 a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).
%F A379515 a(n)/A379516(n) = E * log(n) + F + O(log(n)^(5/3) * log(log(n))^(4/3) / n^u), where u > 0, E = A308041 * (2/(A323482 + 1/2) - 1) = 0.10259754363391420806..., and F is a constant.
%e A379515 Fractions begin with 1, 2/3, 11/12, 43/60, 53/60, 4/5, 37/40, 293/360, 329/360, 103/120, 113/120, 107/120, ...
%t A379515 usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]
%o A379515 (PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
%o A379515 list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(numerator(s), ", "))};
%Y A379515 Cf. A034448, A064609, A308041, A323482, A370898, A379513, A379516 (denominators).
%K A379515 nonn,easy,frac
%O A379515 1,2
%A A379515 _Amiram Eldar_, Dec 23 2024