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A358911
Number of integer compositions of n whose parts all have the same number of prime factors, counted with multiplicity.
9
1, 1, 2, 2, 3, 4, 4, 7, 9, 12, 20, 21, 39, 49, 79, 109, 161, 236, 345, 512, 752, 1092, 1628, 2376, 3537, 5171, 7650, 11266, 16634, 24537, 36173, 53377, 78791, 116224, 171598, 253109, 373715, 551434, 814066, 1201466, 1773425, 2617744, 3864050, 5703840, 8419699
OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from Lucas A. Brown)
Lucas A. Brown, Python program.
EXAMPLE
The a(1) = 1 through a(8) = 9 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (23) (33) (25) (35)
(1111) (32) (222) (52) (44)
(11111) (111111) (223) (53)
(232) (233)
(322) (323)
(1111111) (332)
(2222)
(11111111)
MAPLE
b:= proc(n, i) option remember; uses numtheory; `if`(n=0, 1, add(
(t-> `if`(i<0 or i=t, b(n-j, t), 0))(bigomega(j)), j=1..n))
end:
a:= n-> b(n, -1):
seq(a(n), n=0..44); # Alois P. Heinz, Feb 12 2024
MATHEMATICA
Table[Length[Select[Join @@ Permutations/@IntegerPartitions[n], SameQ@@PrimeOmega/@#&]], {n, 0, 10}]
CROSSREFS
The case of partitions is A319169, ranked by A320324.
The weakly decreasing version is A358335, strictly A358901.
For sequences of partitions see A358905.
A001222 counts prime factors, distinct A001221.
A011782 counts compositions.
A358902 = compositions with weakly decreasing A001221, strictly A358903.
A358909 = partitions with weakly decreasing A001222, complement A358910.
Sequence in context: A056099 A084848 A342330 * A153937 A357710 A242971
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 11 2022
EXTENSIONS
a(21) and beyond from Lucas A. Brown, Dec 15 2022
STATUS
approved