A080084 Number of prime factors in the factorial of the n-th prime, counted with multiplicity.

1, 2, 5, 8, 16, 20, 29, 33, 41, 56, 60, 76, 85, 89, 98, 114, 129, 134, 151, 160, 166, 180, 192, 207, 229, 240, 244, 254, 260, 271, 308, 321, 338, 342, 369, 374, 391, 409, 418, 435, 451, 457, 484, 492, 502, 507, 541, 572, 585, 590, 601, 616, 623, 653, 674, 689

Offset: 1

Author

Paul D. Hanna, Jan 26 2003

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Crossrefs

Cf. A080085, A080086, A080087.

Programs

Maple

b:= proc(n) option remember; `if`(n=1, 0,
      b(n-1)+numtheory[bigomega](n))
    end:
a:= n-> b(ithprime(n)):
seq(a(n), n=1..60);  # Alois P. Heinz, Jul 05 2019

Mathematica

PrimeOmega[#!]&/@Prime[Range[60]] (* Harvey P. Dale, Nov 09 2011 *)

Formula

a(n) = Sum_{m=1..n} Sum_{k=1..L} floor( p_n /(p_m)^k ), where L = ceiling( log(p_n)/log(p_m) ).