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A358911 Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.

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