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

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

Author

Gus Wiseman, Dec 11 2022

Keywords

• nonn

Examples

			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)
		

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.

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.

Cf. A056239, A063834, A064573, A218482, A279787, A300335, A319066, A319071, A358831, A358908.

Programs

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}]

Extensions

a(21) and beyond from Lucas A. Brown, Dec 15 2022