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%I #8 Dec 28 2020 09:54:15
%S 9,25,30,49,63,70,75,84,100,121,147,154,165,169,175,189,196,198,210,
%T 220,250,264,273,280,286,289,325,343,351,361,363,364,385,390,441,442,
%U 462,468,484,490,495,507,520,525,529,550,561,588,594,595,616,624,637,646
%N Heinz numbers of non-graphical, multigraphical integer partitions of even numbers.
%C An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph, and multigraphical if it comprises the multiset of vertex-degrees of some multigraph.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DegreeSequence.html">Degree Sequence.</a>
%H Gus Wiseman, <a href="/A339741/a339741_1.txt">Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.</a>
%F Equals A320924 /\ A339618.
%F Equals A320924 \ A320922.
%e The sequence of terms together with their prime indices begins:
%e 9: {2,2} 189: {2,2,2,4} 363: {2,5,5}
%e 25: {3,3} 196: {1,1,4,4} 364: {1,1,4,6}
%e 30: {1,2,3} 198: {1,2,2,5} 385: {3,4,5}
%e 49: {4,4} 210: {1,2,3,4} 390: {1,2,3,6}
%e 63: {2,2,4} 220: {1,1,3,5} 441: {2,2,4,4}
%e 70: {1,3,4} 250: {1,3,3,3} 442: {1,6,7}
%e 75: {2,3,3} 264: {1,1,1,2,5} 462: {1,2,4,5}
%e 84: {1,1,2,4} 273: {2,4,6} 468: {1,1,2,2,6}
%e 100: {1,1,3,3} 280: {1,1,1,3,4} 484: {1,1,5,5}
%e 121: {5,5} 286: {1,5,6} 490: {1,3,4,4}
%e 147: {2,4,4} 289: {7,7} 495: {2,2,3,5}
%e 154: {1,4,5} 325: {3,3,6} 507: {2,6,6}
%e 165: {2,3,5} 343: {4,4,4} 520: {1,1,1,3,6}
%e 169: {6,6} 351: {2,2,2,6} 525: {2,3,3,4}
%e 175: {3,3,4} 361: {8,8} 529: {9,9}
%e For example, a complete list of all multigraphs with degrees (4,2,2,2) is:
%e {{1,2},{1,2},{1,3},{1,4},{3,4}}
%e {{1,2},{1,3},{1,3},{1,4},{2,4}}
%e {{1,2},{1,3},{1,4},{1,4},{2,3}}
%e Since none of these is strict, i.e., a graph, the Heinz number 189 is in the sequence.
%t strr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strr[n/d],Min@@#>=d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],EvenQ[Length[nrmptn[#]]]&& Select[strr[Times@@Prime/@nrmptn[#]],UnsameQ@@#&]=={}&&strr[Times@@Prime/@nrmptn[#]]!={}&]
%Y See link for additional cross references.
%Y Distinct prime shadows (images under A181819) of A340017.
%Y A000070 counts non-multigraphical partitions (A339620).
%Y A000569 counts graphical partitions (A320922).
%Y A027187 counts partitions of even length (A028260).
%Y A058696 counts partitions of even numbers (A300061).
%Y A096373 cannot be partitioned into strict pairs.
%Y A209816 counts multigraphical partitions (A320924).
%Y A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
%Y A320893 can be partitioned into distinct pairs but not into strict pairs.
%Y A339560 can be partitioned into distinct strict pairs.
%Y A339617 counts non-graphical partitions of 2n (A339618).
%Y A339659 counts graphical partitions of 2n into k parts.
%Y Cf. A004251, A006129, A007717, A056239, A095268, A112798, A181821, A305936, A318284, A339559.
%K nonn
%O 1,1
%A _Gus Wiseman_, Dec 27 2020
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