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A319169
Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.
22
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
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2500 (first 101 terms from Chai Wah Wu)
EXAMPLE
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
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 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 10 2018
EXTENSIONS
a(51)-a(58) from Chai Wah Wu, Nov 12 2018
STATUS
approved