A358910 Number of integer partitions of n whose parts do not have weakly decreasing numbers of prime factors (A001222).

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 11, 19, 25, 41, 56, 84, 113, 164, 218, 306, 401, 547, 711, 949, 1218, 1599, 2034, 2625, 3310, 4224, 5283, 6664, 8271, 10336, 12747, 15791, 19343, 23791, 28979, 35398, 42887, 52073, 62779, 75804, 90967, 109291, 130605

Offset: 0

Author

Gus Wiseman, Dec 09 2022

Keywords

• nonn

Examples

			The a(9) = 1 through a(14) = 11 partitions:
  (54)  (541)  (74)    (543)    (76)      (554)
               (542)   (741)    (544)     (743)
               (5411)  (5421)   (742)     (761)
                       (54111)  (5422)    (5432)
                                (5431)    (5441)
                                (7411)    (7421)
                                (54211)   (54221)
                                (541111)  (54311)
                                          (74111)
                                          (542111)
                                          (5411111)
		

Crossrefs

For sequences of partitions see A141199, compositions A218482.

The case of equality is A319169, for compositions A358911.

The complement is counted by A358909.

A001222 counts prime factors, distinct A001221.

A063834 counts twice-partitions.

Cf. A056239, A300335, A320324, A358831, A358902, A358903, A358908.

Programs

Mathematica

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