%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