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%I #9 Dec 24 2024 07:29:28
%S 1,4,19,107,39,61,259,817,853,97,301,307,2209,187,2279,39583,121129,
%T 122557,124699,126127,509863,171541,173921,526523,6930479,6983519,
%U 7063079,7118771,7193027,802663,405199,13495327,1131701,30726097,123670153,622026437,11910394103
%N Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).
%H Amiram Eldar, <a href="/A379513/b379513.txt">Table of n, a(n) for n = 1..1000</a>
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
%H V. Sita Ramaiah and D. Suryanarayana, <a href="https://web.archive.org/web/20200803214209/http://www.insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005bab_1334.pdf">Sums of reciprocals of some multiplicative functions - II</a>, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
%H 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.
%H Rimer Zurita, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zurita/zur3.html">Generalized Alternating Sums of Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.
%F a(n) = numerator(Sum_{k=1..n} 1/A034448(k)).
%F a(n)/A379514(n) = B * log(n) + D + O(log(n)^(5/3) * log(log(n))^(4/3) / n), where B = A308041, D = B * (gamma + A1 - A2), gamma = A001620, A1 = Sum_{p prime} ((p*C(p)*log(p)/(p-1)) * Sum_{k>=1} (k/(p^k*(p^(k+1)+1)))), A2 = Sum_{p prime} ((C(p)*log(p)/p^2) * Sum_{k>=0} (1/(p^k*(p^(k+1)+1)))), and C(p) = 1 - (p/(p-1)) * Sum_{k>=1} (1/(p^k*(p^(k+1)+1))) (Sita Ramaiah and Suryanarayana, 1980).
%e Fractions begin with 1, 4/3, 19/12, 107/60, 39/20, 61/30, 259/120, 817/360, 853/360, 97/40, 301/120, 307/120, ...
%t usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]
%o (PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
%o list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(numerator(s), ", "))};
%Y Cf. A034448, A064609, A370898, A379514 (denominators), A379515.
%Y Cf. A001620, A308041.
%K nonn,easy,frac
%O 1,2
%A _Amiram Eldar_, Dec 23 2024