A050345 Number of ways to factor n into distinct factors with one level of parentheses.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 6, 1, 3, 3, 4, 1, 6, 1, 6, 3, 3, 1, 13, 1, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 15, 1, 3, 3, 13, 1, 12, 1, 6, 6, 3, 1, 25, 1, 6, 3, 6, 1, 13, 3, 13, 3, 3, 1, 31, 1, 3, 6, 12, 3, 12, 1, 6, 3, 12, 1, 37, 1, 3, 6, 6, 3, 12, 1, 25, 4, 3, 1, 31, 3, 3, 3, 13, 1, 31, 3, 6, 3, 3

Offset

1,6

Comments

Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.

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..2519

Formula

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A045778(n).

a(n) = A050346(A025487^(-1)(A046523(n))), where A025487^(-1) is the inverse with A025487^(-1)(A025487(n))=n. - R. J. Mathar, May 25 2017

a(n) = A050346(A101296(n)). - Antti Karttunen, May 25 2017

Example

12 = (12) = (6*2) = (6)*(2) = (4*3) = (4)*(3) = (3*2)*(2).

Crossrefs

Cf. A045778, A050346-A050350. a(p^k)=A050342. a(A002110)=A000258.

Sequence in context: A254578 A348610 A296120 * A348611 A070039 A227991

Adjacent sequences: A050342 A050343 A050344 * A050346 A050347 A050348

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved