A358903 Number of integer partitions of n whose parts have all different numbers of distinct prime factors (A001221).

1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 7, 8, 7, 9, 10, 10, 10, 9, 11, 15, 14, 13, 15, 14, 14, 17, 16, 17, 17, 16, 16, 17, 17, 21, 26, 24, 23, 25, 27, 29, 32, 31, 29, 36, 36, 35, 37, 37, 42, 49, 45, 44, 50, 49, 50, 58, 55, 55, 58, 56, 58, 66, 62, 65, 75

Offset: 0

Author

Gus Wiseman, Dec 07 2022

Keywords

• nonn

Examples

			The a(15) = 8 partitions are: (15), (14,1), (12,3), (12,2,1), (10,5), (10,4,1), (6,9), (8,6,1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 101 terms from Lucas A. Brown)
• Lucas A. Brown, Python program.

Crossrefs

Counting prime factors with multiplicity gives A358901.

The weakly decreasing version is A358902, with multiplicity A358335.

A001222 counts prime factors, distinct A001221.

A116608 counts partitions by sum and number of distinct parts.

A358836 counts multiset partitions with all distinct block sizes.

Cf. A046660, A071625, A129519, A141199, A319169, A358831, A358909, A358911.

Programs

Maple

p:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      add((t-> `if`(t b(n$2):
seq(a(n), n=0..68);  # Alois P. Heinz, Feb 14 2024

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@PrimeNu/@#&]],{n,0,30}]

Extensions

a(56) and beyond from Lucas A. Brown, Dec 14 2022