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A025281
a(n) = sopfr(n!), where sopfr = A001414 is the integer log.
8
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.
Sequence in context: A373799 A266899 A112265 * A160663 A024201 A110443
KEYWORD
nonn
STATUS
approved