A050379 Number of ordered factorizations of n into members of A050376.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 6, 1, 5, 1, 5, 2, 2, 1, 10, 2, 2, 3, 5, 1, 6, 1, 10, 2, 2, 2, 14, 1, 2, 2, 10, 1, 6, 1, 5, 5, 2, 1, 22, 2, 5, 2, 5, 1, 10, 2, 10, 2, 2, 1, 18, 1, 2, 5, 18, 2, 6, 1, 5, 2, 6, 1, 32, 1, 2, 5, 5, 2, 6, 1, 22, 6, 2, 1, 18, 2, 2, 2, 10, 1, 18, 2, 5, 2, 2, 2

Offset: 1

Author

Christian G. Bower, Nov 15 1999

Keywords

• nonn

Comments

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A002033, A050376, A050377, A050378, A050380.

Cf. also A000142, A002110, A023359.

Programs

Maple

read(transforms) :
L := [1] :
for n from 2 to 100  do
    if isA050376(n) then
        L := [op(L),-1] ;
    else
        L := [op(L),0] ;
    end if;
end do :
a050379 := DIRICHLETi(L) ; # R. J. Mathar, May 26 2017

PARI

A064547(n) = {my(f = factor(n)[, 2]); sum(k=1, #f, hammingweight(f[k])); } \\ Michel Marcus, Feb 10 2016
isA050376(n) = ((1==omega(n)) && (1==A064547(n))); \\ Checking that omega(n) is 1 is just an optimization here.
A050379(n) = if(1==n,n,sumdiv(n,d,if(dA050376(n/d)*A050379(d),0))); \\ Antti Karttunen, Oct 20 2017

Formula

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A050376.

a(p^k) = A023359(k), for any prime p.

a(A002110(n)) = A000142(n) = n!.

a(n) = A050380(A101296(n)). - R. J. Mathar, May 26 2017