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

Author

Eric W. Weisstein, Jul 01 2004

Keywords

• nonn
• frac

Examples

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

References

• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 167.

Links

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

Crossrefs

Cf. A096625.

Programs

Mathematica

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

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

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

a(n)/A096625(n) = Sum_{p prime <= n} HarmonicNumber(floor(log_p n)). - Ammar Khatab, Jan 25 2025