A256012 Number of partitions of n into distinct parts that are not squarefree.

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 1, 1, 0, 3, 2, 1, 0, 4, 3, 1, 2, 5, 4, 2, 2, 6, 5, 3, 2, 9, 7, 4, 4, 11, 8, 5, 5, 13, 13, 7, 7, 17, 17, 9, 9, 22, 20, 15, 12, 27, 26, 19, 15, 33, 33, 23, 23, 41, 41, 30, 29, 49, 51, 39, 35, 65, 63, 50, 47, 79

Offset: 0

Author

Reinhard Zumkeller, Jun 01 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for n > 23.

Examples

			First nonsquarefree numbers: 4,8,9,12,16,18,20,24,25,27,28, ...  hence
a(20) = #{20, 16+4, 12+8} = 3;
a(21) = #{12+9, 9+8+4} = 2;
a(22) = #{18+4} = 1;
a(23) = #{ } = 0;
a(24) = #{24, 20+4, 16+8, 12+8+4} = 4;
a(25) = #{25, 16+9, 12+9+4} = 3.
		

Links

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

Crossrefs

Cf. A013929, A114374, A087188.

Programs

Haskell

a256012 = p a013929_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Maple

with(numtheory):
b:= proc(n, i) option remember;
      `if`(i*(i+1)/2n or issqrfree(i), 0, b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Jun 02 2015

Mathematica

b[n_, i_] := b[n, i] = If[i*(i+1)/2n || SquareFreeQ[i], 0, b[n-i, i-1]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 22 2015, after Alois P. Heinz *)

Formula

G.f.: Product_{k>=1} (1 + x^k)/(1 + mu(k)^2*x^k), where mu(k) is the Moebius function (A008683). - Ilya Gutkovskiy, Dec 30 2016