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

0,4

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.

Example

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

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<i, b(n-j, t), 0))(p(j)), j=1..n)))
    end:
a:= n-> 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}]

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.

Sequence in context: A025780 A199121 A109697 * A103373 A038539 A275891

Adjacent sequences: A358900 A358901 A358902 * A358904 A358905 A358906

Keyword

nonn

Author

Gus Wiseman, Dec 07 2022

Extensions

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

Status

approved