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Number of prime factors in the factorial of the n-th prime, counted with multiplicity.
4

%I #13 Jul 05 2019 16:58:33

%S 1,2,5,8,16,20,29,33,41,56,60,76,85,89,98,114,129,134,151,160,166,180,

%T 192,207,229,240,244,254,260,271,308,321,338,342,369,374,391,409,418,

%U 435,451,457,484,492,502,507,541,572,585,590,601,616,623,653,674,689

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

%H Alois P. Heinz, <a href="/A080084/b080084.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)

%F 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) ).

%p b:= proc(n) option remember; `if`(n=1, 0,

%p b(n-1)+numtheory[bigomega](n))

%p end:

%p a:= n-> b(ithprime(n)):

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Jul 05 2019

%t PrimeOmega[#!]&/@Prime[Range[60]] (* _Harvey P. Dale_, Nov 09 2011 *)

%Y Cf. A080085, A080086, A080087.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jan 26 2003