OFFSET
1,2
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.
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).
EXAMPLE
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, ...
MATHEMATICA
uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); uphi[1] = 1; Numerator[Accumulate[Table[1/uphi[n], {n, 1, 50}]]]
PROG
(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), ", "))};
CROSSREFS
KEYWORD
nonn,easy,frac
AUTHOR
Amiram Eldar, Dec 24 2024
STATUS
approved