A200727 Number of partitions of n such that the number of parts is not divisible by the greatest part.

0, 1, 1, 3, 4, 8, 9, 16, 22, 33, 42, 61, 79, 110, 143, 192, 246, 325, 411, 535, 676, 865, 1081, 1371, 1704, 2136, 2642, 3283, 4035, 4979, 6082, 7453, 9067, 11043, 13365, 16197, 19516, 23531, 28239, 33894, 40513, 48425, 57667, 68661, 81497, 96679, 114370

Offset: 1

Author

Alois P. Heinz, Nov 21 2011

Keywords

• nonn

Comments

Also number of partitions of n such that the greatest part is not divisible by the number of parts. Equivalence can be shown using Ferrers-Young diagrams.

Examples

			The number of parts is not divisible by the greatest part:
a(5) = 4: [1,2,2], [2,3], [1,4], [5];
a(6) = 8: [1,1,1,1,2], [2,2,2], [1,1,1,3], [3,3], [1,1,4], [2,4], [1,5], [6].
The greatest part is not divisible by the number of parts:
a(5) = 4: [1,1,1,1,1], [1,1,1,2], [1,2,2], [2,3];
a(6) = 8: [1,1,1,1,1,1], [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [3,3], [1,1,4], [1,5].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A199884, A199885.

Programs

Maple

b:= proc(n, j, t) option remember;
      add(b(n-i, i, t+1), i=j..iquo(n, 2))+
      `if`(irem(t, n)>0, 1, 0)
    end:
a:= n-> b(n, 1, 1):
seq(a(n), n=1..50);

Mathematica

b[n_, j_, t_] := b[n, j, t] = Sum[b[n-i, i, t+1], {i, j, Quotient[n, 2]}] + If[Mod[t, n]>0, 1, 0]; a[n_] := b[n, 1, 1]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Feb 05 2017, translated from Maple *)