A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).

1, 1, 2, 3, 5, 8, 12, 19, 29, 44, 68, 100, 153, 227, 342, 509, 759, 1129, 1678, 2492, 3699, 5477, 8121, 12015, 17795, 26313, 38924, 57541, 85065, 125712, 185758, 274431, 405420, 598815, 884465, 1306165, 1928943, 2848360, 4205979, 6210289, 9169540

Offset: 0

Author

Gus Wiseman, Dec 05 2022

Keywords

• nonn

Examples

			The a(0) = 1 through a(6) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (23)     (33)
                 (111)  (31)    (32)     (42)
                        (211)   (41)     (51)
                        (1111)  (221)    (222)
                                (311)    (231)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Lucas A. Brown)
• Lucas A. Brown, Python program.

Crossrefs

For lengths of partitions see A141199, compositions A218482.

The strictly decreasing case is A358901.

The version not counting multiplicity is A358902, strict A358903.

The case of partitions is A358909, complement A358910.

The case of equality is A358911, partitions A319169.

A001222 counts prime factors, distinct A001221.

A011782 counts compositions.

A063834 counts twice-partitions.

Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.

Programs

Mathematica

Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022