login
A050363
Number of ordered factorizations into prime powers greater than 1.
7
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
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
KEYWORD
nonn
AUTHOR
Christian G. Bower, Oct 15 1999
STATUS
approved