A096624 - OEIS

%I #33 Jan 25 2025 13:39:15

%S 0,1,2,5,7,7,9,29,16,16,19,19,22,22,22,91,103,103,115,115,115,115,127,

%T 127,133,133,137,137,149,149,161,817,817,817,817,817,877,877,877,877,

%U 937,937,997,997,997,997,1057,1057,1087,1087,1087,1087,1147,1147,1147

%N Numerators of the Riemann prime counting function.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html">Riemann Prime Counting Function</a>

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

%F (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

%F a(n)/A096625(n) = Sum_{p prime <= n} HarmonicNumber(floor(log_p n)). - _Ammar Khatab_, Jan 25 2025

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

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

%o (PARI) a(n) = numerator(sum(k=1, n, if (p=isprimepower(k), 1/p))); \\ _Michel Marcus_, Jan 07 2019

%o (PARI) a(n) = numerator(sum(k=1, logint(n, 2), primepi(sqrtnint(n, k))/k)); \\ _Daniel Suteu_, Jan 07 2019

%Y Cf. A096625.

%K nonn,frac,changed

%O 1,3

%A _Eric W. Weisstein_, Jul 01 2004