This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339655 #14 Feb 16 2025 08:34:01
%S A339655 0,0,1,3,7,14,28,51,91,156,260,425,680,1068,1654,2524,3802,5668,8350,
%T A339655 12190,17634,25306,36011,50902,71441,99642
%N A339655 Number of non-loop-graphical integer partitions of 2n.
%C A339655 An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
%C A339655 The following are equivalent characteristics for any positive integer n:
%C A339655 (1) the prime factors of n can be partitioned into distinct pairs;
%C A339655 (2) n can be factored into distinct semiprimes;
%C A339655 (3) the prime signature of n is loop-graphical.
%H A339655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339655 A058696(n) = a(n) + A339656(n).
%e A339655 The a(2) = 1 through a(5) = 14 partitions (A = 10):
%e A339655 (4) (6) (8) (A)
%e A339655 (4,2) (4,4) (5,5)
%e A339655 (5,1) (5,3) (6,4)
%e A339655 (6,2) (7,3)
%e A339655 (7,1) (8,2)
%e A339655 (5,2,1) (9,1)
%e A339655 (6,1,1) (5,3,2)
%e A339655 (5,4,1)
%e A339655 (6,2,2)
%e A339655 (6,3,1)
%e A339655 (7,2,1)
%e A339655 (8,1,1)
%e A339655 (6,2,1,1)
%e A339655 (7,1,1,1)
%e A339655 For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
%e A339655 {{1,1},{1,1},{1,2},{2,2},{3,3}}
%e A339655 {{1,1},{1,1},{1,2},{2,3},{2,3}}
%e A339655 {{1,1},{1,1},{1,3},{2,2},{2,3}}
%e A339655 {{1,1},{1,2},{1,2},{1,2},{3,3}}
%e A339655 {{1,1},{1,2},{1,2},{1,3},{2,3}}
%e A339655 {{1,1},{1,2},{1,3},{1,3},{2,2}}
%e A339655 {{1,2},{1,2},{1,2},{1,3},{1,3}},
%e A339655 but since none of these is a loop-graph (because they are not strict), y is counted under a(5).
%t A339655 spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
%t A339655 mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
%t A339655 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A339655 Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]
%Y A339655 A001358 lists semiprimes, with squarefree case A006881.
%Y A339655 A006125 counts labeled graphs, with covering case A006129.
%Y A339655 A062740 counts labeled connected loop-graphs.
%Y A339655 A101048 counts partitions into semiprimes.
%Y A339655 A320461 ranks normal loop-graphs.
%Y A339655 A322661 counts covering loop-graphs.
%Y A339655 A320655 counts factorizations into semiprimes.
%Y A339655 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339655 - A058696 counts partitions of 2n (A300061).
%Y A339655 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339655 - A209816 counts multigraphical partitions (A320924).
%Y A339655 - A339655 (this sequence) counts non-loop-graphical partitions of 2n (A339657).
%Y A339655 - A339656 counts loop-graphical partitions (A339658).
%Y A339655 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339655 - A000569 counts graphical partitions (A320922).
%Y A339655 The following count partitions of even length and give their Heinz numbers:
%Y A339655 - A027187 has no additional conditions (A028260).
%Y A339655 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339655 - A338914 can be partitioned into strict pairs (A320911).
%Y A339655 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339655 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339655 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339655 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339655 Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.
%K A339655 nonn,more
%O A339655 0,4
%A A339655 _Gus Wiseman_, Dec 14 2020
%E A339655 a(7)-a(25) from _Andrew Howroyd_, Jan 10 2024