cp's OEIS Frontend

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

%I A358335 #16 Feb 12 2024 17:24:53
%S A358335 1,1,2,3,5,8,12,19,29,44,68,100,153,227,342,509,759,1129,1678,2492,
%T A358335 3699,5477,8121,12015,17795,26313,38924,57541,85065,125712,185758,
%U A358335 274431,405420,598815,884465,1306165,1928943,2848360,4205979,6210289,9169540
%N A358335 Number of integer compositions of n whose parts have weakly decreasing numbers of prime factors (with multiplicity).
%H A358335 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 A358335 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358335.py">Python program</a>.
%e A358335 The a(0) = 1 through a(6) = 12 compositions:
%e A358335   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A358335            (11)  (21)   (22)    (23)     (33)
%e A358335                  (111)  (31)    (32)     (42)
%e A358335                         (211)   (41)     (51)
%e A358335                         (1111)  (221)    (222)
%e A358335                                 (311)    (231)
%e A358335                                 (2111)   (321)
%e A358335                                 (11111)  (411)
%e A358335                                          (2211)
%e A358335                                          (3111)
%e A358335                                          (21111)
%e A358335                                          (111111)
%t A358335 Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeOmega/@#&]],{n,0,10}]
%Y A358335 For lengths of partitions see A141199, compositions A218482.
%Y A358335 The strictly decreasing case is A358901.
%Y A358335 The version not counting multiplicity is A358902, strict A358903.
%Y A358335 The case of partitions is A358909, complement A358910.
%Y A358335 The case of equality is A358911, partitions A319169.
%Y A358335 A001222 counts prime factors, distinct A001221.
%Y A358335 A011782 counts compositions.
%Y A358335 A063834 counts twice-partitions.
%Y A358335 Cf. A056239, A300335, A319071, A320324, A358831, A358904, A358908.
%K A358335 nonn
%O A358335 0,3
%A A358335 _Gus Wiseman_, Dec 05 2022
%E A358335 a(21) and beyond from _Lucas A. Brown_, Dec 15 2022