A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.

1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 7, 11, 11, 14, 15, 20, 19, 26, 27, 34, 35, 43, 45, 59, 60, 72, 77, 94, 98, 118, 125, 148, 158, 184, 198, 233, 245, 282, 308, 353, 374, 428, 464, 525, 566, 635, 686, 779, 832, 930, 1005, 1123, 1208, 1345, 1451, 1609, 1732, 1912

Offset: 0

Author

Gus Wiseman, Oct 10 2018

Keywords

• nonn

Examples

			The a(1) = 1 through a(9) = 6 integer partitions:
  1  2   3    4     5      6       7        8         9
     11  111  22    32     33      52       44        72
              1111  11111  222     322      53        333
                           111111  1111111  332       522
                                            2222      3222
                                            11111111  111111111
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2500 (first 101 terms from Chai Wah Wu)

Crossrefs

Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.

Programs

Maple

b:= proc(n, i, f) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, f)+(o-> `if`(f in {0, o}, b(n-i, min(i, n-i),
     `if`(f=0, o, f)), 0))(numtheory[bigomega](i))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..75);  # Alois P. Heinz, Dec 15 2018

Mathematica

Table[Length[Select[IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,30}]
(* Second program: *)
b[n_, i_, f_] := b[n, i, f] = If[n == 0, 1, If[i < 1, 0,
     b[n, i-1, f] + Function[o, If[f == 0 || f == o, b[n-i, Min[i, n-i],
     If[f == 0, o, f]], 0]][PrimeOmega[i]]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 75] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Extensions

a(51)-a(58) from Chai Wah Wu, Nov 12 2018