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

0,3

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.

Example

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)

Mathematica

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

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.

Sequence in context: A124062 A274199 A099823 * A240523 A023436 A024567

Adjacent sequences: A358332 A358333 A358334 * A358336 A358337 A358338

Keyword

nonn

Author

Gus Wiseman, Dec 05 2022

Extensions

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

Status

approved