A025281 a(n) = sopfr(n!), where sopfr = A001414 is the integer log.

0, 0, 2, 5, 9, 14, 19, 26, 32, 38, 45, 56, 63, 76, 85, 93, 101, 118, 126, 145, 154, 164, 177, 200, 209, 219, 234, 243, 254, 283, 293, 324, 334, 348, 367, 379, 389, 426, 447, 463, 474, 515, 527, 570, 585, 596, 621, 668, 679, 693, 705, 725, 742, 795, 806, 822, 835, 857, 888

Offset: 0

Author

Wouter Meeussen

Keywords

• nonn

References

• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 144.

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms n = 0..1000 from T. D. Noe)
• Krishnaswami Alladi and Paul Erdős, On an additive arithmetic function, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294; alternative link.

Crossrefs

Partial sums of A001414.

Cf. A000142, A072691.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, 0,
      a(n-1)+add(i[1]*i[2], i=ifactors(n)[2]))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 09 2021

Mathematica

sopfr[n_] := Plus @@ Times @@@ FactorInteger@ n; a[n_] := a[n] = a[n - 1] + sopfr[n]; a[0] = a[1] = 0; Array[a, 59, 0] (* Robert G. Wilson v, May 18 2015 *)

PARI

for(n=1,100,print1(sum(k=1,n,sum(i=1,omega(k), component(component(factor(k),1),i)*component(component(factor(k),2),i))),","))

Python

from sympy import factorial, factorint
def A025281(n): return sum(p*e for p, e in factorint(factorial(n)).items()) # Chai Wah Wu, Apr 09 2021

Formula

a(n) = A001414(A000142(n)).

From Benoit Cloitre, Apr 14 2002: (Start)

a(0)=0; for n>0, a(n) = Sum_{k=1..n} A001414(k).

Asymptotic formula: a(n) ~ (Pi^2/12)*n^2/log(n). [Proven by Alladi and Erdős (1977). - Amiram Eldar, Mar 04 2021]

(End)