A050334 Number of ordered factorizations of n into numbers with an odd number of prime divisors (prime factors counted with multiplicity).

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 8, 1, 2, 2, 4, 1, 7, 1, 5, 2, 2, 2, 10, 1, 2, 2, 8, 1, 7, 1, 4, 4, 2, 1, 15, 1, 4, 2, 4, 1, 8, 2, 8, 2, 2, 1, 18, 1, 2, 4, 8, 2, 7, 1, 4, 2, 7, 1, 23, 1, 2, 4, 4, 2, 7, 1, 15, 3, 2, 1, 18, 2, 2, 2, 8, 1, 18, 2, 4, 2, 2, 2, 28, 1, 4, 4

Offset

1,6

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

R. J. Mathar, Table of n, a(n) for n = 1..10000

Formula

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of A026424 (essentially A066829).

a(p^k) = A000045(k).

a(A002110(k)) = A006154(k).

a(n) = A050335(A101296(n)). - R. J. Mathar, May 26 2017

Example

From R. J. Mathar, May 25 2017: (Start)
a(p) = 1: factorizations p.
a(p^2) = 1: factorizations p*p.
a(p^3) = 2: factorizations p^3, p*p*p.
a(p^4) = 3: factorizations p^3*p, p*p^3, p*p*p*p.
a(p^5) = 5: factorizations p^5, p^3*p*p, p*p^3*p, p*p*p^3, p*p*p*p*p.
a(p*q) = 2: factorizations p*q, q*p. (End)

Maple

read(transforms):
A066829m := proc(n)
        if n = 1 or isA026424(n) then
                1;
        else
                0;
        end if;
end proc:
[1, seq(-A066829m(n), n=2..10000)] ;
DIRICHLETi(%) ; # R. J. Mathar, May 25 2017

Crossrefs

Cf. A002033, A026424, A050333.

Sequence in context: A264440 A303386 A295636 * A342084 A295279 A316784

Adjacent sequences: A050331 A050332 A050333 * A050335 A050336 A050337

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved