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A319169
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Number of integer partitions of n whose parts all have the same number of prime factors, counted with multiplicity.
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21
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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
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OFFSET
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0,3
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LINKS
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EXAMPLE
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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
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MAPLE
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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):
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MATHEMATICA
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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];
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CROSSREFS
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Cf. A000607, A001222, A003963, A064573, A279787, A305551, A319056, A319066, A319071, A320322, A320324.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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