A096624 Numerators of the Riemann prime counting function.

0, 1, 2, 5, 7, 7, 9, 29, 16, 16, 19, 19, 22, 22, 22, 91, 103, 103, 115, 115, 115, 115, 127, 127, 133, 133, 137, 137, 149, 149, 161, 817, 817, 817, 817, 817, 877, 877, 877, 877, 937, 937, 997, 997, 997, 997, 1057, 1057, 1087, 1087, 1087, 1087, 1147, 1147, 1147

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Riemann Prime Counting Function

Formula

Let Sk{f(k)}= Sum_{k>=2}f(k), then the g.f. of A096624/A096625 can be written as

(1/1)*Sa{(x^a)/(1-x)} - (1/2)*Sa{ Sb{ (x^(a*b))/(1-x)}} + (1/3)*Sa{ Sb{ Sc{ (x^(a*b*c))/(1-x)}}} - (1/4)*Sa{ Sb{ Sc{ Sd{ (x^(a*b*c*d))/(1-x)}}}} + ... . - Mats Granvik, Apr 06 2011

Example

0, 1, 2, 5/2, 7/2, 7/2, 9/2, 29/6, 16/3, 16/3, 19/3, ...

Mathematica

Table[Sum[PrimePi[x^(1/k)]/k, {k, Log2[x]}], {x, 100}] // Numerator (* Eric W. Weisstein, Jan 09 2019 *)

Prog

(PARI) a(n) = numerator(sum(k=1, n, if (p=isprimepower(k), 1/p))); \\ Michel Marcus, Jan 07 2019
(PARI) a(n) = numerator(sum(k=1, logint(n, 2), primepi(sqrtnint(n, k))/k)); \\ Daniel Suteu, Jan 07 2019

Crossrefs

Cf. A096625.

Sequence in context: A131688 A226213 A199590 * A145378 A069887 A254340

Adjacent sequences: A096621 A096622 A096623 * A096625 A096626 A096627

Keyword

nonn,frac

Author

Eric W. Weisstein, Jul 01 2004

Status

approved