A379517 Numerators of the partial sums of the reciprocals of the unitary totient function (A047994).
1, 2, 5, 17, 37, 43, 15, 109, 225, 239, 1223, 3809, 1293, 4019, 1031, 209, 1693, 1735, 5261, 5345, 5429, 27649, 306659, 310619, 312929, 317549, 4155857, 4195897, 603091, 615961, 619393, 19304143, 19463731, 1228951, 9898103, 4982299, 1251116, 2524397, 10164083
Offset: 1
Examples
Fractions begin with 1, 2, 5/2, 17/6, 37/12, 43/12, 15/4, 109/28, 225/56, 239/56, 1223/280, 3809/840, ...
Links
- Amiram Eldar, Table of n, a(n) for n = 1..1000
- Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018. See p. 52.
- 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 4.10, pp. 30-31.
- Rimer Zurita, Generalized Alternating Sums of Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 23 (2020), Article 20.10.4. See section 4.5, pp. 16-17.
Programs
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Mathematica
uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[1/uphi[n], {n, 1, 50}]]] -
PARI
uphi(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 / uphi(k); print1(numerator(s), ", "))};
Formula
a(n) = numerator(Sum_{k=1..n} 1/A047994(k)).
a(n)/A379518(n) = L * log(n) + M + O(log(n)^(5/3)/n), where L = A327837, M = L * (gamma - B + A1 + A2), gamma = A001620, B = Sum_{p prime} (1-1/p) * log(p) * Sum_{k>=1} k/(p^k*(p^k-1)) / A(p), A1 = Sum_{p prime} log(p)/(p^2*(p-1)*A(p)), A2 = Sum_{p prime} ((A*(p)(p)*log(p)/p^2), A(p) = 1 + (1-1/p) * Sum_{k>=1} 1/(p^k*(p^k-1)), and A*(p) = Sum_{k>=1} 1/(p^k*p^(k+1)-1)*A(p)) (Sita Ramaiah and Suryanarayana, 1980).