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

0,3

Comments

See A062051 for partitions into distinct 3-smooth numbers.

Links

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

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000041, A003586.

Cf. A117220, A117221.

Cf. A062051, A023893, A131995.

Sequence in context: A304405 A347446 A097071 * A058641 A212253 A237831

Adjacent sequences: A105417 A105418 A105419 * A105421 A105422 A105423

Keyword

nonn

Author

Reinhard Zumkeller, Apr 07 2005

Status

approved