A105420 Number of partitions of n into 3-smooth parts.

1, 1, 2, 3, 5, 6, 10, 12, 18, 23, 31, 38, 53, 63, 82, 100, 128, 152, 194, 228, 284, 336, 410, 478, 586, 678, 814, 947, 1127, 1296, 1539, 1761, 2070, 2372, 2764, 3146, 3667, 4153, 4796, 5437, 6249, 7044, 8080, 9080, 10358, 11636, 13208, 14778, 16762, 18698

Offset: 0

Author

Reinhard Zumkeller, Apr 07 2005

Keywords

• nonn

Comments

See A062051 for partitions into distinct 3-smooth numbers.

Examples

			n=10: there are 11 partitions of 10 with at least one part not of the form 2^i*3^j: 10, 7+3, 7+2+1, 7+1+1+1, 5+5, 5+4+1, 5+3+2, 5+3+1+1, 5+2+2+1, 5+2+1+1+1 and 5+1+1+1+1+1, therefore a(10) = A000041(10) - 11 = 42 - 11 = 31.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 121 terms from Jean-François Alcover)

Crossrefs

Cf. A000041, A003586.

Cf. A117220, A117221.

Cf. A062051, A023893, A131995.

Programs

Mathematica

nmax = 120;
S = Select[Range[nmax], Max[FactorInteger[#][[All, 1]]] <= 3 &];
P[n_] := IntegerPartitions[n, All, TakeWhile[S, # <= n &] ];
a[n_] := a[n] = P[n] // Length;
Table[Print[n, " ", a[n]]; a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 13 2021 *)

Formula

A117222(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006