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

0,3

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.

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)

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 *)

Prog

(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

Crossrefs

Partial sums of A001414.

Cf. A000142, A072691.

Sequence in context: A060139 A266899 A112265 * A160663 A024201 A110443

Adjacent sequences: A025278 A025279 A025280 * A025282 A025283 A025284

Keyword

nonn

Author

Wouter Meeussen

Status

approved