cp's OEIS Frontend

A338915 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.

%I A338915 #24 Feb 16 2025 08:34:01
%S A338915 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,
%T A338915 319,415,507,649,786,983,1198,1499,1797,2234,2673,3303,3952,4826,5753,
%U A338915 6999,8330,10051,11943,14357,16956,20322,23997,28568,33657,39897,46879
%N A338915 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.
%C A338915 The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).
%H A338915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition</a>.
%F A338915 A027187(n) = a(n) + A338916(n).
%e A338915 The a(7) = 1 through a(12) = 12 partitions:
%e A338915   211111  2222      411111    222211      222221      3333
%e A338915           221111    21111111  331111      611111      222222
%e A338915           311111              511111      22211111    441111
%e A338915           11111111            22111111    32111111    711111
%e A338915                               31111111    41111111    22221111
%e A338915                               1111111111  2111111111  32211111
%e A338915                                                       33111111
%e A338915                                                       42111111
%e A338915                                                       51111111
%e A338915                                                       2211111111
%e A338915                                                       3111111111
%e A338915                                                       111111111111
%e A338915 For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
%e A338915   {{1,1},{1,1},{1,2},{2,3}}
%e A338915   {{1,1},{1,1},{1,3},{2,2}}
%e A338915   {{1,1},{1,2},{1,2},{1,3}}
%e A338915 None of these is strict, so y is counted under a(12).
%t A338915 smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
%t A338915 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]
%Y A338915 The Heinz numbers of these partitions are A320892.
%Y A338915 The complement in even-length partitions is A338916.
%Y A338915 A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
%Y A338915 A000569 counts graphical partitions, ranked by A320922.
%Y A338915 A001358 lists semiprimes, with squarefree case A006881.
%Y A338915 A058696 counts partitions of even numbers, ranked by A300061.
%Y A338915 A209816 counts multigraphical partitions, ranked by A320924.
%Y A338915 A320655 counts factorizations into semiprimes.
%Y A338915 A322353 counts factorizations into distinct semiprimes.
%Y A338915 A339617 counts non-graphical partitions of 2n, ranked by A339618.
%Y A338915 A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
%Y A338915 A339656 counts loop-graphical partitions, ranked by A339658.
%Y A338915 The following count partitions of even length and give their Heinz numbers:
%Y A338915 - A027187 has no additional conditions (A028260).
%Y A338915 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A338915 - A338914 can be partitioned into strict pairs (A320911).
%Y A338915 - A338916 can be partitioned into distinct pairs (A320912).
%Y A338915 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A338915 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A338915 Cf. A001055, A007717, A025065, A320656, A320732, A320893, A338898, A338902.
%K A338915 nonn
%O A338915 0,9
%A A338915 _Gus Wiseman_, Dec 10 2020
%E A338915 More terms from _Jinyuan Wang_, Feb 14 2025