A028415 Numerator of Sum_{k=1..n} 1/phi(k).

1, 2, 5, 3, 13, 15, 47, 25, 13, 55, 281, 74, 301, 311, 637, 163, 1319, 453, 4117, 4207, 4267, 4339, 48089, 49079, 9895, 10027, 10115, 10247, 72125, 73511, 369403, 93217, 9391, 75821, 76283, 77207, 77515, 78131, 78593, 39643, 49727, 100609, 100939, 25408, 204419

Offset: 1

Author

N. J. A. Sloane, Jun 28 2002

Keywords

• nonn
• frac

Examples

			1, 2, 5/2, 3, 13/4, 15/4, 47/12, 25/6, 13/3, 55/12, 281/60, 74/15, ...
		

References

• József Sándor, Dragoslav S. Mitrinovic, and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Section I.27, page 29.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• R. Sitaramachandrarao, On an error term of Landau - II, The Rocky Mountain Journal of Mathematics, Vol. 15, No. 2 (1985), pp. 579-588.
• Eric Weisstein's World of Mathematics, Totient Summatory Function.

Crossrefs

Cf. A000010, A048049 (denominators).

Cf. A001620, A085609, A082695.

Programs

Maple

map(numer, ListTools:-PartialSums(map(1/numtheory:-phi, [$1..10000]))); # Robert Israel, Apr 16 2019

Mathematica

Numerator[Table[Sum[1/EulerPhi[k],{k,n}],{n,50}]] (* Harvey P. Dale, Aug 24 2012 *)

PARI

a(n) = numerator(sum(k=1, n, 1/eulerphi(k))); \\ Michel Marcus, Sep 18 2022

Formula

a(n)/A048049(n) = c * (log(n) + gamma - s) + O(log(n)^(2/3)/n), where c = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and s = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Sitaramachandrarao, 1985). - Amiram Eldar, Sep 18 2022