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%I #11 Dec 18 2020 07:58:40
%S 1,0,1,1,2,3,5,6,8,12,16,21,28,37,49,64,80,104,135,169,216,268,341,
%T 420,527,654,809,991,1218,1488,1828,2213,2687,3262,3934,4754,5702,
%U 6849,8200,9819,11693
%N Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.
%C The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A027187(n) = a(n) + A338915(n).
%e The a(2) = 1 through a(10) = 16 partitions:
%e (11) (21) (22) (32) (33) (43) (44) (54) (55)
%e (31) (41) (42) (52) (53) (63) (64)
%e (2111) (51) (61) (62) (72) (73)
%e (2211) (2221) (71) (81) (82)
%e (3111) (3211) (3221) (3222) (91)
%e (4111) (3311) (3321) (3322)
%e (4211) (4221) (3331)
%e (5111) (4311) (4222)
%e (5211) (4321)
%e (6111) (4411)
%e (222111) (5221)
%e (321111) (5311)
%e (6211)
%e (7111)
%e (322111)
%e (421111)
%e 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).
%t stfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
%t Table[Length[Select[IntegerPartitions[n],stfs[Times@@Prime/@#]!={}&]],{n,0,20}]
%Y A320912 gives the Heinz numbers of these partitions.
%Y A338915 counts the complement in even-length partitions.
%Y A339563 counts factorizations of the same type.
%Y A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A000569 counts graphical partitions, ranked by A320922.
%Y A001358 lists semiprimes, with squarefree case A006881.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A209816 counts multigraphical partitions, ranked by A320924.
%Y A320655 counts factorizations into semiprimes.
%Y A322353 counts factorizations into distinct semiprimes.
%Y A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y A339656 counts loop-graphical partitions, ranked by A339658.
%Y The following count partitions of even length and give their Heinz numbers:
%Y - A027187 has no additional conditions (A028260).
%Y - A096373 cannot be partitioned into strict pairs (A320891).
%Y - A338914 can be partitioned into strict pairs (A320911).
%Y - A338915 cannot be partitioned into distinct pairs (A320892).
%Y - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y - A339560 can be partitioned into distinct strict pairs (A339561).
%Y Cf. A001055, A007717, A025065, A320656, A320732, A320893, A320921, A338898, A338902, A339564.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Dec 10 2020
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