A081401 Pseudologarithm (A056239) of n!: a(n) = A056239(A000142(n)).

0, 1, 3, 5, 8, 11, 15, 18, 22, 26, 31, 35, 41, 46, 51, 55, 62, 67, 75, 80, 86, 92, 101, 106, 112, 119, 125, 131, 141, 147, 158, 163, 170, 178, 185, 191, 203, 212, 220, 226, 239, 246, 260, 267, 274, 284, 299, 305, 313, 320, 329, 337, 353, 360, 368, 375, 385, 396, 413

Offset: 1

Author

Labos Elemer, Mar 31 2003

Keywords

• nonn

Comments

As A056239 is fully additive sequence, this sequence gives its partial sums. - Antti Karttunen, Jun 28 2020

Examples

			n=8: 8! = 40320 = 2*2*2*2*2*2*2*3*3*5*7, p-indices = {1,2,3,4}, exponents = {7,2,1,1}; a(8) = 1*7 + 2*2 + 3*1 + 4*1 = 7 + 4 + 3 + 4 = 18.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000142, A056239.

Cf. also A335860.

Programs

Maple

a:= n-> add (numtheory[pi](i[1])*i[2], i=ifactors(n!)[2]):
seq (a(n), n=1..100);  # Alois P. Heinz, Aug 09 2012

Mathematica

Array[Total[FactorInteger[#!] /. {p_, c_} /; p > 0 :> PrimePi[p] c] &, 59] (* Michael De Vlieger, Jun 26 2020 *)

Formula

a(n) = Sum(k*e(k)) where k runs through indices of prime factors of n!, while e(k) is the exponent of the corresponding prime factor.