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A338916
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Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.
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20
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1, 0, 1, 1, 2, 3, 5, 6, 8, 12, 16, 21, 28, 37, 49, 64, 80, 104, 135, 169, 216, 268, 341, 420, 527, 654, 809, 991, 1218, 1488, 1828, 2213, 2687, 3262, 3934, 4754, 5702, 6849, 8200, 9819, 11693
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OFFSET
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0,5
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COMMENTS
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The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).
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LINKS
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FORMULA
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EXAMPLE
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The a(2) = 1 through a(10) = 16 partitions:
(11) (21) (22) (32) (33) (43) (44) (54) (55)
(31) (41) (42) (52) (53) (63) (64)
(2111) (51) (61) (62) (72) (73)
(2211) (2221) (71) (81) (82)
(3111) (3211) (3221) (3222) (91)
(4111) (3311) (3321) (3322)
(4211) (4221) (3331)
(5111) (4311) (4222)
(5211) (4321)
(6111) (4411)
(222111) (5221)
(321111) (5311)
(6211)
(7111)
(322111)
(421111)
For example, the partition (4,2,1,1,1,1) can be partitioned into {{1,1},{1,2},{1,4}} so is counted under a(10).
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MATHEMATICA
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stfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[stfs[n/d], Min@@#>d&]], {d, Select[Rest[Divisors[n]], PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n], stfs[Times@@Prime/@#]!={}&]], {n, 0, 20}]
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CROSSREFS
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A320912 gives the Heinz numbers of these partitions.
A338915 counts the complement in even-length partitions.
A339563 counts factorizations of the same type.
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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