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A339658 Heinz numbers of loop-graphical partitions (of even numbers).

%I A339658 #12 Feb 16 2025 08:34:01
%S A339658 1,3,4,9,10,12,16,25,27,28,30,36,40,48,63,64,70,75,81,84,88,90,100,
%T A339658 108,112,120,144,147,160,175,189,192,196,198,208,210,220,225,243,250,
%U A339658 252,256,264,270,280,300,324,336,343,352,360,400,432,441,448,462,468,480
%N A339658 Heinz numbers of loop-graphical partitions (of even numbers).
%C A339658 Equals the image of A181819 applied to the set of terms of A320912.
%C A339658 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%C A339658 A 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. Loop-graphical partitions are counted by A339656.
%C A339658 The following are equivalent characteristics for any positive integer n:
%C A339658 (1) the prime factors of n can be partitioned into distinct pairs;
%C A339658 (2) n can be factored into distinct semiprimes;
%C A339658 (3) the prime signature of n is loop-graphical.
%H A339658 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339658 A300061 = A339657 \/ A339658.
%e A339658 The sequence of terms > 1 together with their prime indices begins:
%e A339658       3: {2}               70: {1,3,4}          192: {1,1,1,1,1,1,2}
%e A339658       4: {1,1}             75: {2,3,3}          196: {1,1,4,4}
%e A339658       9: {2,2}             81: {2,2,2,2}        198: {1,2,2,5}
%e A339658      10: {1,3}             84: {1,1,2,4}        208: {1,1,1,1,6}
%e A339658      12: {1,1,2}           88: {1,1,1,5}        210: {1,2,3,4}
%e A339658      16: {1,1,1,1}         90: {1,2,2,3}        220: {1,1,3,5}
%e A339658      25: {3,3}            100: {1,1,3,3}        225: {2,2,3,3}
%e A339658      27: {2,2,2}          108: {1,1,2,2,2}      243: {2,2,2,2,2}
%e A339658      28: {1,1,4}          112: {1,1,1,1,4}      250: {1,3,3,3}
%e A339658      30: {1,2,3}          120: {1,1,1,2,3}      252: {1,1,2,2,4}
%e A339658      36: {1,1,2,2}        144: {1,1,1,1,2,2}    256: {1,1,1,1,1,1,1,1}
%e A339658      40: {1,1,1,3}        147: {2,4,4}          264: {1,1,1,2,5}
%e A339658      48: {1,1,1,1,2}      160: {1,1,1,1,1,3}    270: {1,2,2,2,3}
%e A339658      63: {2,2,4}          175: {3,3,4}          280: {1,1,1,3,4}
%e A339658      64: {1,1,1,1,1,1}    189: {2,2,2,4}        300: {1,1,2,3,3}
%e A339658 For example, the four loop-graphs with degrees y = (3,1,1,1) are:
%e A339658   {{1,1},{1,2},{3,4}}
%e A339658   {{1,1},{1,3},{2,4}}
%e A339658   {{1,1},{1,4},{2,3}}
%e A339658   {{1,2},{1,3},{1,4}},
%e A339658 so the Heinz number 40 is in the sequence. On the other hand, the three loop-multigraphs with degrees y = (4,4) are
%e A339658   {{1,1},{1,1},{2,2},{2,2}}
%e A339658   {{1,1},{1,2},{1,2},{2,2}}
%e A339658   {{1,2},{1,2},{1,2},{1,2}},
%e A339658 but none of these is a loop-graph, so the Heinz number 49 is not in the sequence.
%t A339658 spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
%t A339658 mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
%t A339658 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A339658 Select[Range[25],Select[mpsbin[nrmptn[#]],UnsameQ@@#&]!={}&]
%Y A339658 A320912 has these prime shadows (see A181819).
%Y A339658 A339656 counts these partitions.
%Y A339658 A339657 ranks the complement, counted by A339655.
%Y A339658 A001358 lists semiprimes, with squarefree case A006881.
%Y A339658 A101048 counts partitions into semiprimes.
%Y A339658 A320655 counts factorizations into semiprimes.
%Y A339658 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339658 - A058696 counts partitions of 2n (A300061).
%Y A339658 - A209816 counts multigraphical partitions (A320924).
%Y A339658 - A000569 counts graphical partitions (A320922).
%Y A339658 The following count partitions of even length and give their Heinz numbers:
%Y A339658 - A027187 has no additional conditions (A028260).
%Y A339658 - A338914 can be partitioned into strict pairs (A320911).
%Y A339658 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339658 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339658 Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A320892, A338898, A338912, A338913, A339112.
%K A339658 nonn
%O A339658 1,2
%A A339658 _Gus Wiseman_, Dec 18 2020