A050374 Number of ordered factorizations of n into composite factors.

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 0, 1, 0, 3, 1, 1, 1, 4, 0, 1, 1, 3, 0, 1, 0, 1, 1, 1, 0, 5, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 1, 5, 1, 1, 0, 1, 1, 1, 0, 7, 0, 1, 1, 1, 1, 1, 0, 5, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5, 1, 1, 1, 1, 1, 10, 0, 1, 1, 4, 0, 1

Offset

1,16

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).

The Dirichlet inverse is given by A005171, all but the first term in A005171 turned negative. - R. J. Mathar, Jul 15 2010

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n

Formula

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

a(n) = A050375(A101296(n)). - R. J. Mathar, May 26 2017

For n >= 1, a(p^n) = A000045(n-1), for any prime p.

For n >= 0, a(A002110(n)) = A032032(n).

Maple

read(transforms):
[1, seq(-A005171(n), n=2..100)] ;
a050374 := DIRICHLETi(%) ; # R. J. Mathar, May 26 2017

Prog

(PARI) A050374(n) = if(1==n, n, sumdiv(n, d, if(d<n, !isprime(n/d)*A050374(d), 0))); \\ Antti Karttunen, Oct 20 2017

Crossrefs

Cf. A002033, A002808, A005171, A050370-A050375, A000045, A002110, A032032.

Sequence in context: A167746 A360616 A050370 * A238877 A047886 A239366

Adjacent sequences: A050371 A050372 A050373 * A050375 A050376 A050377

Keyword

nonn

Author

Christian G. Bower, Nov 15 1999

Status

approved