A050358 Number of ordered factorizations of n with 3 levels of parentheses.

1, 1, 1, 5, 1, 9, 1, 25, 5, 9, 1, 65, 1, 9, 9, 125, 1, 65, 1, 65, 9, 9, 1, 425, 5, 9, 25, 65, 1, 121, 1, 625, 9, 9, 9, 605, 1, 9, 9, 425, 1, 121, 1, 65, 65, 9, 1, 2625, 5, 65, 9, 65, 1, 425, 9, 425, 9, 9, 1, 1145, 1, 9, 65, 3125, 9, 121, 1, 65, 9, 121, 1, 4825, 1, 9, 65, 65, 9, 121

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 given by A050356, turning all but the first element of A050356 negative. - R. J. Mathar, Jul 15 2010

Links

R. J. Mathar, Table of n, a(n) for n = 1..899

Formula

Dirichlet g.f.: (4-3*zeta(s))/(5-4*zeta(s)).

a(n) = A050359(A101296(n)). - R. J. Mathar, May 26 2017

Sum_{k=1..n} a(k) ~ -n^r / (16*r*Zeta'(r)), where r = 2.7884327053324956670606046076818023223650950899573090550836329583345... is the root of the equation Zeta(r) = 5/4. - Vaclav Kotesovec, Feb 02 2019

Example

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

Crossrefs

Cf. A002033, A050351-A050359. a(p^k)=5^(k-1). a(A002110)=A050353.

Sequence in context: A147085 A348980 A046580 * A147319 A147326 A336048

Adjacent sequences: A050355 A050356 A050357 * A050359 A050360 A050361

Keyword

nonn

Author

Christian G. Bower, Oct 15 1999

Status

approved