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%I #17 Feb 12 2024 19:02:44
%S 1,1,2,2,3,4,4,7,9,12,20,21,39,49,79,109,161,236,345,512,752,1092,
%T 1628,2376,3537,5171,7650,11266,16634,24537,36173,53377,78791,116224,
%U 171598,253109,373715,551434,814066,1201466,1773425,2617744,3864050,5703840,8419699
%N Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.
%H Alois P. Heinz, <a href="/A358911/b358911.txt">Table of n, a(n) for n = 0..4000</a> (first 101 terms from Lucas A. Brown)
%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358911.py">Python program</a>.
%e The a(1) = 1 through a(8) = 9 compositions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (23) (33) (25) (35)
%e (1111) (32) (222) (52) (44)
%e (11111) (111111) (223) (53)
%e (232) (233)
%e (322) (323)
%e (1111111) (332)
%e (2222)
%e (11111111)
%p b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
%p (t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
%p end:
%p a:= n-> b(n, -1):
%p seq(a(n), n=0..44); # _Alois P. Heinz_, Feb 12 2024
%t Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,0,10}]
%Y The case of partitions is A319169, ranked by A320324.
%Y The weakly decreasing version is A358335, strictly A358901.
%Y For sequences of partitions see A358905.
%Y A001222 counts prime factors, distinct A001221.
%Y A011782 counts compositions.
%Y A358902 = compositions with weakly decreasing A001221, strictly A358903.
%Y A358909 = partitions with weakly decreasing A001222, complement A358910.
%Y Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 11 2022
%E a(21) and beyond from _Lucas A. Brown_, Dec 15 2022
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