A177329 Number of factors in the representation of n! as a product of distinct terms of A050376.

1, 2, 3, 4, 3, 4, 6, 6, 4, 5, 7, 8, 9, 10, 11, 12, 8, 9, 9, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 19, 21, 17, 16, 15, 16, 17, 18, 19, 20, 22, 23, 21, 21, 21, 22, 23, 22, 23, 25, 22, 23, 22, 24, 26, 28, 28, 29, 27, 28, 29, 30, 32, 34, 30, 31, 31, 28, 27, 28, 29, 30, 31, 33, 31, 31, 30

Offset: 2

Author

Vladimir Shevelev, May 06 2010

Keywords

• nonn

References

• Vladimir S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 [Russian].

Links

• Chai Wah Wu, Table of n, a(n) for n = 2..10000 (terms 2..1000 from Amiram Eldar)
• Simon Litsyn and Vladimir Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Crossrefs

Cf. A000120, A001358, A050292, A050376, A064380, A064547, A176472, A176509, A176525.

Programs

Maple

read("transforms") ; A064547 := proc(n) f := ifactors(n)[2] ; a := 0 ; for p in f do a := a+wt(op(2,p)) ; end do: a ; end proc:
A177329 := proc(n) A064547(n!) ; end proc: seq(A177329(n),n=2..80) ; # R. J. Mathar, May 28 2010

Mathematica

f[p_, e_] := DigitCount[e, 2, 1]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n!]; Array[a, 100, 2] (* Amiram Eldar, Aug 24 2024 *)

PARI

a(n) = vecsum(apply(x -> hammingweight(x), factor(n!)[,2])); \\ Amiram Eldar, Aug 24 2024

Python

from collections import Counter
from sympy import factorint
def A177329(n): return sum(map(int.bit_count,sum((Counter(factorint(i)) for i in range(2,n+1)),start=Counter()).values())) # Chai Wah Wu, Jul 18 2024

Formula

a(n) = Sum_{i} A000120(e_i), where n! = Product_{i} p_i^e_i is the prime factorization of n!.

a(n) = A064547(n!). - R. J. Mathar, May 28 2010

Extensions

a(20)=10 inserted by Vladimir Shevelev, May 08 2010

Terms from a(14) onwards replaced according to the formula - R. J. Mathar, May 28 2010