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

1,4

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.

Links

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

Example

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].

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 *)

Crossrefs

Cf. A199884, A199885.

Sequence in context: A254877 A193351 A243985 * A096185 A137640 A165362

Adjacent sequences: A200724 A200725 A200726 * A200728 A200729 A200730

Keyword

nonn

Author

Alois P. Heinz, Nov 21 2011

Status

approved