%I #16 Feb 12 2024 17:24:53

%S 1,1,2,3,5,8,12,19,29,44,68,100,153,227,342,509,759,1129,1678,2492,

%T 3699,5477,8121,12015,17795,26313,38924,57541,85065,125712,185758,

%U 274431,405420,598815,884465,1306165,1928943,2848360,4205979,6210289,9169540

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

%H Alois P. Heinz, <a href="/A358335/b358335.txt">Table of n, a(n) for n = 0..2000</a> (first 101 terms from Lucas A. Brown)

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358335.py">Python program</a>.

%e The a(0) = 1 through a(6) = 12 compositions:

%e () (1) (2) (3) (4) (5) (6)

%e (11) (21) (22) (23) (33)

%e (111) (31) (32) (42)

%e (211) (41) (51)

%e (1111) (221) (222)

%e (311) (231)

%e (2111) (321)

%e (11111) (411)

%e (2211)

%e (3111)

%e (21111)

%e (111111)

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

%Y For lengths of partitions see A141199, compositions A218482.

%Y The strictly decreasing case is A358901.

%Y The version not counting multiplicity is A358902, strict A358903.

%Y The case of partitions is A358909, complement A358910.

%Y The case of equality is A358911, partitions A319169.

%Y A001222 counts prime factors, distinct A001221.

%Y A011782 counts compositions.

%Y A063834 counts twice-partitions.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Dec 05 2022

%E a(21) and beyond from _Lucas A. Brown_, Dec 15 2022