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

0,12

Links

Table of n, a(n) for n=0..50.

Example

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)

Mathematica

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

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.

Sequence in context: A271723 A212545 A357878 * A212546 A212547 A212548

Adjacent sequences: A358907 A358908 A358909 * A358911 A358912 A358913

Keyword

nonn

Author

Gus Wiseman, Dec 09 2022

Status

approved