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

1,4

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

Links

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

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

Example

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)

Maple

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

Crossrefs

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

Sequence in context: A077191 A317545 A319002 * A308063 A166974 A292504

Adjacent sequences: A050360 A050361 A050362 * A050364 A050365 A050366

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved