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

0,13

Comments

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

Links

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

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

Example

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.

Maple

with(numtheory):
b:= proc(n, i) option remember;
      `if`(i*(i+1)/2<n, 0, `if`(n=0, 1, b(n, i-1)+
      `if`(i>n 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)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n || 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 *)

Prog

(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

Crossrefs

Cf. A013929, A114374, A087188.

Sequence in context: A249808 A258453 A025874 * A332040 A263251 A318370

Adjacent sequences: A256009 A256010 A256011 * A256013 A256014 A256015

Keyword

nonn

Author

Reinhard Zumkeller, Jun 01 2015

Status

approved