A025281 - OEIS

%I #53 Dec 27 2023 11:55:10

%S 0,0,2,5,9,14,19,26,32,38,45,56,63,76,85,93,101,118,126,145,154,164,

%T 177,200,209,219,234,243,254,283,293,324,334,348,367,379,389,426,447,

%U 463,474,515,527,570,585,596,621,668,679,693,705,725,742,795,806,822,835,857,888

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

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

%H Chai Wah Wu, <a href="/A025281/b025281.txt">Table of n, a(n) for n = 0..10000</a> (terms n = 0..1000 from T. D. Noe)

%H Krishnaswami Alladi and Paul Erdős, <a href="https://projecteuclid.org/euclid.pjm/1102811427">On an additive arithmetic function</a>, Pacific Journal of Mathematics, Vol. 71, No. 2 (1977), pp. 275-294; <a href="https://msp.org/pjm/1977/71-2/pjm-v71-n2-p01-s.pdf">alternative link</a>.

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

%F From _Benoit Cloitre_, Apr 14 2002: (Start)

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

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

%F (End)

%p a:= proc(n) option remember; `if`(n<2, 0,

%p       a(n-1)+add(i[1]*i[2], i=ifactors(n)[2]))

%p     end:

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Apr 09 2021

%t 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 *)

%o (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))),","))

%o (Python)

%o from sympy import factorial, factorint

%o def A025281(n): return sum(p*e for p, e in factorint(factorial(n)).items()) # _Chai Wah Wu_, Apr 09 2021

%Y Partial sums of A001414.

%Y Cf. A000142, A072691.

%K nonn

%O 0,3

%A _Wouter Meeussen_