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A338915
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Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.
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23
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0, 0, 0, 0, 1, 0, 1, 1, 4, 2, 6, 6, 12, 12, 20, 22, 38, 42, 60, 73, 101, 124, 164, 203, 266, 319, 415, 507, 649, 786, 983, 1198, 1499, 1797, 2234, 2673, 3303, 3952, 4826, 5753, 6999
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OFFSET
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0,9
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COMMENTS
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The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).
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LINKS
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FORMULA
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EXAMPLE
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The a(7) = 1 through a(12) = 12 partitions:
211111 2222 411111 222211 222221 3333
221111 21111111 331111 611111 222222
311111 511111 22211111 441111
11111111 22111111 32111111 711111
31111111 41111111 22221111
1111111111 2111111111 32211111
33111111
42111111
51111111
2211111111
3111111111
111111111111
For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
{{1,1},{1,1},{1,2},{2,3}}
{{1,1},{1,1},{1,3},{2,2}}
{{1,1},{1,2},{1,2},{1,3}}
None of these is strict, so y is counted under a(12).
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MATHEMATICA
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smcs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[smcs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]], {n, 0, 10}]
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CROSSREFS
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The Heinz numbers of these partitions are A320892.
The complement in even-length partitions is A338916.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
The following count partitions of even length and give their Heinz numbers:
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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