A050328 Number of ordered factorizations of n into squarefree numbers > 1.

1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 5, 1, 3, 3, 1, 1, 5, 1, 5, 3, 3, 1, 7, 1, 3, 1, 5, 1, 13, 1, 1, 3, 3, 3, 13, 1, 3, 3, 7, 1, 13, 1, 5, 5, 3, 1, 9, 1, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 31, 1, 3, 5, 1, 3, 13, 1, 5, 3, 13, 1, 25, 1, 3, 5, 5, 3, 13, 1, 9, 1, 3, 1, 31, 3, 3, 3, 7, 1, 31, 3, 5, 3, 3, 3, 11, 1, 5

Offset: 1

Author

Christian G. Bower, Oct 15 1999

Keywords

• nonn

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

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000670, A000961, A002033, A002110, A005117, A008966, A050329, A114006, A206778.

Programs

Haskell

import Data.List (genericIndex)
a050328 n = genericIndex a050328_list (n-1)
a050328_list = f 1 where
   f x = (if x == 1 then 1 else
         sum $ map (a050328 . (div x)) $ tail $ a206778_row x) : f (x + 1)
-- Reinhard Zumkeller, May 03 2013

Mathematica

a[n_]:=If[n==1, n, Sum[If[(dIndranil Ghosh, May 27 2017 *)

PARI

A050328(n) = if(1==n,n,sumdiv(n,d,if((dA050328(d),0))); \\ Antti Karttunen, May 27 2017

Formula

Dirichlet g.f.: 1/(1-B(s)) where B(s) is D.g.f. of characteristic function of squarefree numbers > 1.

a(A000961(n)) = 1.

a(A002110(n)) = A000670(n).

a(n) = Sum_{d divides n, dA008966(n/d)*a(d). - Vladeta Jovovic, Sep 25 2002, corrected by Antti Karttunen, May 27 2017

G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} mu(k)^2*A(x^k). - Ilya Gutkovskiy, May 10 2019