A073576 Number of partitions of n into squarefree parts.

1, 1, 2, 3, 4, 6, 9, 12, 16, 21, 28, 36, 47, 60, 76, 96, 120, 150, 185, 228, 280, 342, 416, 504, 608, 731, 877, 1048, 1249, 1484, 1759, 2079, 2452, 2885, 3387, 3968, 4640, 5413, 6304, 7328, 8504, 9852, 11395, 13159, 15172, 17468, 20082, 23056, 26434, 30267

Offset

0,3

Comments

Euler transform of the absolute values of A008683. - Tilman Neumann, Dec 13 2008

Euler transform of A008966. - Vaclav Kotesovec, Mar 31 2018

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: 1/Product_{k>0} (1-x^A005117(k)).

a(n) = 1/n*Sum_{k=1..n} A048250(k)*a(n-k).

a(n) = A000041(n) - A114374(n) - A117395(n), n>0. - Reinhard Zumkeller, Mar 11 2006

G.f.: 1 + Sum_{i>=1} mu(i)^2*x^i / Product_{j=1..i} (1 - mu(j)^2*x^j). - Ilya Gutkovskiy, Jun 05 2017

a(n) ~ exp(2*sqrt(n)) / (4*Pi^(3/2)*n^(1/4)). - Vaclav Kotesovec, Mar 24 2018

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      abs(mobius(d)), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 05 2015

Mathematica

Table[Length[Select[Boole /@ Thread /@ SquareFreeQ /@ IntegerPartitions[n], FreeQ[#, 0] &]], {n, 48}] (* Jayanta Basu, Jul 02 2013 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*Abs[MoebiusMu[d]], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 10 2015, after Alois P. Heinz *)
nmax = 60; CoefficientList[Series[Exp[Sum[Sum[Abs[MoebiusMu[k]] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

Prog

(Haskell)
a073576 = p a005117_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Jun 01 2015

Crossrefs

Cf. A058647.

Cf. A087188.

Cf. A225244.

Cf. A114374.

Sequence in context: A060729 A229169 A155510 * A186115 A187020 A058647

Adjacent sequences: A073573 A073574 A073575 * A073577 A073578 A073579

Keyword

nonn

Author

Vladeta Jovovic, Aug 27 2002

Status

approved