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A379513
Numerators of the partial sums of the reciprocals of the sum of unitary divisors function (A034448).
0
1, 4, 19, 107, 39, 61, 259, 817, 853, 97, 301, 307, 2209, 187, 2279, 39583, 121129, 122557, 124699, 126127, 509863, 171541, 173921, 526523, 6930479, 6983519, 7063079, 7118771, 7193027, 802663, 405199, 13495327, 1131701, 30726097, 123670153, 622026437, 11910394103
OFFSET
1,2
LINKS
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 51.
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions - II, Indian J. Pure Appl. Math., Vol. 11 (1980), pp. 1334-1355.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.
Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.3, pp. 12-15.
FORMULA
a(n) = numerator(Sum_{k=1..n} 1/A034448(k)).
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).
EXAMPLE
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, ...
MATHEMATICA
usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Numerator[Accumulate[Table[1/usigma[n], {n, 1, 50}]]]
PROG
(PARI) usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += 1 / usigma(k); print1(numerator(s), ", "))};
CROSSREFS
Cf. A034448, A064609, A370898, A379514 (denominators), A379515.
Sequence in context: A369109 A082030 A348802 * A052751 A367284 A249934
KEYWORD
nonn,easy,frac,new
AUTHOR
Amiram Eldar, Dec 23 2024
STATUS
approved