cp's OEIS Frontend

A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).

%I A358902 #19 Feb 14 2024 09:56:49
%S A358902 1,1,2,3,5,8,13,21,33,53,84,134,213,338,536,850,1349,2136,3389,5367,
%T A358902 8509,13480,21362,33843,53624,84957,134600,213251,337850,535251,
%U A358902 847987,1343440,2128372,3371895,5341977,8463051,13407689,21241181,33651507,53312538,84460690
%N A358902 Number of integer compositions of n whose parts have weakly decreasing numbers of distinct prime factors (A001221).
%H A358902 Alois P. Heinz, <a href="/A358902/b358902.txt">Table of n, a(n) for n = 0..5004</a> (first 101 terms from Lucas A. Brown)
%H A358902 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A358902.py">Python program</a>.
%e A358902 The a(0) = 1 through a(6) = 13 compositions:
%e A358902   ()  (1)  (2)   (3)    (4)     (5)      (6)
%e A358902            (11)  (21)   (22)    (23)     (24)
%e A358902                  (111)  (31)    (32)     (33)
%e A358902                         (211)   (41)     (42)
%e A358902                         (1111)  (221)    (51)
%e A358902                                 (311)    (222)
%e A358902                                 (2111)   (231)
%e A358902                                 (11111)  (321)
%e A358902                                          (411)
%e A358902                                          (2211)
%e A358902                                          (3111)
%e A358902                                          (21111)
%e A358902                                          (111111)
%p A358902 p:= proc(n) option remember; nops(ifactors(n)[2]) end:
%p A358902 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A358902       add((t-> `if`(t<=i, b(n-j, t), 0))(p(j)), j=1..n)))
%p A358902     end:
%p A358902 a:= n-> b(n$2):
%p A358902 seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 14 2024
%t A358902 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],GreaterEqual@@PrimeNu/@#&]],{n,0,10}]
%Y A358902 For lengths of partitions see A141199, compositions A218482.
%Y A358902 The strictly decreasing case is A358903.
%Y A358902 A001222 counts prime factors, distinct A001221.
%Y A358902 A011782 counts compositions.
%Y A358902 A116608 counts partitions by sum and number of distinct parts.
%Y A358902 A334028 counts distinct parts in standard compositions.
%Y A358902 A358836 counts multiset partitions with all distinct block sizes.
%Y A358902 Cf. A046660, A056239, A071625, A129519, A300335, A358831, A358908, A358911.
%K A358902 nonn
%O A358902 0,3
%A A358902 _Gus Wiseman_, Dec 07 2022
%E A358902 a(21) and beyond from _Lucas A. Brown_, Dec 15 2022