A358909 Number of integer partitions of n whose parts have weakly decreasing numbers of prime factors (A001222).

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 53, 73, 93, 124, 157, 206, 256, 329, 406, 514, 628, 784, 949, 1174, 1411, 1725, 2061, 2500, 2966, 3570, 4217, 5039, 5919, 7027, 8219, 9706, 11301, 13268, 15394, 17995, 20792, 24195, 27863, 32288, 37061, 42779, 48950, 56306

Offset: 0

Author

Gus Wiseman, Dec 09 2022

Keywords

• nonn

Comments

First differs from A000041 at a(9) = 29, A000041(9) = 30, the difference coming from the partition (5,4).

Crossrefs

For sequences of partitions see A141199, compositions A218482.

The case of equality is A319169, for compositions A358911.

The case of compositions is A358335, strictly decreasing A358901.

The complement is counted by A358910.

A001222 counts prime factors, distinct A001221.

A011782 counts compositions.

A063834 counts twice-partitions.

Cf. A056239, A300335, A320324, A358831, A358903, A358908.

Programs

Mathematica

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