A050363 Number of ordered factorizations into prime powers greater than 1.

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 5, 1, 2, 2, 8, 1, 5, 1, 5, 2, 2, 1, 12, 2, 2, 4, 5, 1, 6, 1, 16, 2, 2, 2, 14, 1, 2, 2, 12, 1, 6, 1, 5, 5, 2, 1, 28, 2, 5, 2, 5, 1, 12, 2, 12, 2, 2, 1, 18, 1, 2, 5, 32, 2, 6, 1, 5, 2, 6, 1, 37, 1, 2, 5, 5, 2, 6, 1, 28, 8, 2, 1, 18, 2, 2, 2, 12, 1, 18, 2, 5, 2, 2, 2, 64

Offset: 1

Author

Christian G. Bower, Oct 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).

The Dirichlet inverse is in A010055, turning all but the first element in A010055 negative. - R. J. Mathar, Jul 15 2010

Not multiplicative: a(6) =2 <> a(2)*a(3) = 1*1. - R. J. Mathar, May 25 2017

Examples

			From _R. J. Mathar_, May 25 2017: (Start)
a(p^2)  = 2: factorizations p^2, p*p.
a(p^3)  = 4: factorizations p^3, p^2*p, p*p^2, p*p*p.
a(p*q)  = 2: factorizations p*q, q*p.
a(p*q^2)= 5: factorizations p*q^2, q^2*p, p*q*q, q*p*q, q*q*p. (End)
		

Links

• R. J. Mathar, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000961, A050360, A050361, A050362, A050363, A050364, A076277.

Programs

Maple

read(transforms) ;
[1,seq(-A010055(n),n=2..100)] ;
DIRICHLETi(%) ; # R. J. Mathar, May 25 2017

Formula

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

a(p^k) = 2^(k-1).

a(A002110(k)) = k!.

a(n) = A050364(A101296(n)). - R. J. Mathar, May 26 2017

G.f. A(x) satisfies: A(x) = x + Sum_{p prime, k>=1} A(x^(p^k)). - Ilya Gutkovskiy, May 11 2019