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 A339656 #29 Feb 16 2025 08:34:01
%S A339656 1,2,4,8,15,28,49,84,140,229,367,577,895,1368,2064,3080,4547,6642,
%T A339656 9627,13825,19704,27868,39164,54656,75832,104584
%N A339656 Number of loop-graphical integer partitions of 2n.
%C A339656 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 two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.
%C A339656 The following are equivalent characteristics for any positive integer n:
%C A339656 (1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
%C A339656 (2) n can be factored into distinct semiprimes;
%C A339656 (3) the unordered prime signature of n is loop-graphical.
%H A339656 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339656 A058696(n) = a(n) + A339655(n).
%e A339656 The a(0) = 1 through a(4) = 15 partitions:
%e A339656 () (2) (2,2) (3,3) (3,3,2)
%e A339656 (1,1) (3,1) (2,2,2) (4,2,2)
%e A339656 (2,1,1) (3,2,1) (4,3,1)
%e A339656 (1,1,1,1) (4,1,1) (2,2,2,2)
%e A339656 (2,2,1,1) (3,2,2,1)
%e A339656 (3,1,1,1) (3,3,1,1)
%e A339656 (2,1,1,1,1) (4,2,1,1)
%e A339656 (1,1,1,1,1,1) (5,1,1,1)
%e A339656 (2,2,2,1,1)
%e A339656 (3,2,1,1,1)
%e A339656 (4,1,1,1,1)
%e A339656 (2,2,1,1,1,1)
%e A339656 (3,1,1,1,1,1)
%e A339656 (2,1,1,1,1,1,1)
%e A339656 (1,1,1,1,1,1,1,1)
%e A339656 For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
%e A339656 {{1,1},{2,2},{3,4}}
%e A339656 {{1,1},{2,3},{2,4}}
%e A339656 {{1,2},{1,3},{2,4}}
%e A339656 {{1,2},{1,4},{2,3}}
%e A339656 {{1,3},{1,4},{2,2}},
%e A339656 so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
%e A339656 {{1,1},{1,1},{2,2}}
%e A339656 {{1,1},{1,2},{1,2}},
%e A339656 but neither of these is a loop-graph, so z is not counted under a(3).
%t A339656 spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
%t A339656 mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
%t A339656 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A339656 Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]!={}&]],{n,0,5}]
%Y A339656 A339658 ranks these partitions.
%Y A339656 A001358 lists semiprimes, with squarefree case A006881.
%Y A339656 A006125 counts labeled graphs, with covering case A006129.
%Y A339656 A027187 counts partitions of even length, ranked by A028260.
%Y A339656 A062740 counts labeled connected loop-graphs.
%Y A339656 A320461 ranks normal loop-graphs.
%Y A339656 A320655 counts factorizations into semiprimes.
%Y A339656 A322353 counts factorizations into distinct semiprimes.
%Y A339656 A322661 counts covering loop-graphs.
%Y A339656 A339845 counts the same partitions by length, or A339844 with zeros.
%Y A339656 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339656 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339656 - A000569 counts graphical partitions (A320922).
%Y A339656 - A058696 counts partitions of 2n (A300061).
%Y A339656 - A209816 counts multigraphical partitions (A320924).
%Y A339656 - A321728 is conjectured to count non-half-loop-graphical partitions of n.
%Y A339656 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339656 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339656 - A339656 [this sequence] counts loop-graphical partitions (A339658).
%Y A339656 The following count partitions of even length and give their Heinz numbers:
%Y A339656 - A027187 has no additional conditions (A028260).
%Y A339656 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339656 - A338914 can be partitioned into strict pairs (A320911).
%Y A339656 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339656 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339656 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339656 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339656 Cf. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.
%K A339656 nonn,more
%O A339656 0,2
%A A339656 _Gus Wiseman_, Dec 14 2020
%E A339656 a(8)-a(25) from _Andrew Howroyd_, Jan 10 2024