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

%I A319169 #33 May 10 2021 21:05:36
%S A319169 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,
%T A319169 77,94,98,118,125,148,158,184,198,233,245,282,308,353,374,428,464,525,
%U A319169 566,635,686,779,832,930,1005,1123,1208,1345,1451,1609,1732,1912
%N A319169 Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.
%H A319169 Alois P. Heinz, <a href="/A319169/b319169.txt">Table of n, a(n) for n = 0..2500</a> (first 101 terms from Chai Wah Wu)
%e A319169 The a(1) = 1 through a(9) = 6 integer partitions:
%e A319169   1  2   3    4     5      6       7        8         9
%e A319169      11  111  22    32     33      52       44        72
%e A319169               1111  11111  222     322      53        333
%e A319169                            111111  1111111  332       522
%e A319169                                             2222      3222
%e A319169                                             11111111  111111111
%p A319169 b:= proc(n, i, f) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A319169       b(n, i-1, f)+(o-> `if`(f in {0, o}, b(n-i, min(i, n-i),
%p A319169      `if`(f=0, o, f)), 0))(numtheory[bigomega](i))))
%p A319169     end:
%p A319169 a:= n-> b(n$2, 0):
%p A319169 seq(a(n), n=0..75);  # _Alois P. Heinz_, Dec 15 2018
%t A319169 Table[Length[Select[IntegerPartitions[n],SameQ@@PrimeOmega/@#&]],{n,30}]
%t A319169 (* Second program: *)
%t A319169 b[n_, i_, f_] := b[n, i, f] = If[n == 0, 1, If[i < 1, 0,
%t A319169      b[n, i-1, f] + Function[o, If[f == 0 || f == o, b[n-i, Min[i, n-i],
%t A319169      If[f == 0, o, f]], 0]][PrimeOmega[i]]]];
%t A319169 a[n_] := b[n, n, 0];
%t A319169 a /@ Range[0, 75] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y A319169 Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.
%K A319169 nonn
%O A319169 0,3
%A A319169 _Gus Wiseman_, Oct 10 2018
%E A319169 a(51)-a(58) from _Chai Wah Wu_, Nov 12 2018