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%I #10 Dec 20 2020 22:24:32
%S 3,7,10,13,19,21,22,28,29,34,37,39,43,46,52,53,55,57,61,62,66,71,76,
%T 79,82,85,87,88,89,91,94,101,102,107,111,113,115,116,117,118,129,130,
%U 131,133,134,136,138,139,146,148,151,155,156,159,163,166,171,172,173
%N Heinz numbers of non-multigraphical partitions of even numbers.
%C An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
%C The following are equivalent characteristics for any positive integer n:
%C (1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);
%C (2) n can be factored into squarefree semiprimes;
%C (3) the unordered prime signature of n is multigraphical.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F Equals A300061 \ A320924.
%F For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.
%e The sequence of terms together with their prime indices begins:
%e 3: {2} 53: {16} 94: {1,15}
%e 7: {4} 55: {3,5} 101: {26}
%e 10: {1,3} 57: {2,8} 102: {1,2,7}
%e 13: {6} 61: {18} 107: {28}
%e 19: {8} 62: {1,11} 111: {2,12}
%e 21: {2,4} 66: {1,2,5} 113: {30}
%e 22: {1,5} 71: {20} 115: {3,9}
%e 28: {1,1,4} 76: {1,1,8} 116: {1,1,10}
%e 29: {10} 79: {22} 117: {2,2,6}
%e 34: {1,7} 82: {1,13} 118: {1,17}
%e 37: {12} 85: {3,7} 129: {2,14}
%e 39: {2,6} 87: {2,10} 130: {1,3,6}
%e 43: {14} 88: {1,1,1,5} 131: {32}
%e 46: {1,9} 89: {24} 133: {4,8}
%e 52: {1,1,6} 91: {4,6} 134: {1,19}
%e For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
%e {{1,1},{1,1},{1,2},{2,3}}
%e {{1,1},{1,1},{1,3},{2,2}}
%e {{1,1},{1,2},{1,2},{1,3}},
%e but since none of these is a multigraph (they have loops), the Heinz number 66 belongs to the sequence.
%t prpts[m_]:=If[Length[m]==0,{{}},Join@@Table[Prepend[#,ipr]&/@prpts[Fold[DeleteCases[#1,#2,{1},1]&,m,ipr]],{ipr,Select[Subsets[Union[m],{2}],MemberQ[#,m[[1]]]&]}]];
%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[#]]]&&prpts[nrmptn[#]]=={}&]
%Y A000070 counts these partitions.
%Y A300061 is a superset.
%Y A320891 has image under A181819 equal to this set of terms.
%Y A001358 lists semiprimes, with squarefree case A006881.
%Y A002100 counts partitions into squarefree semiprimes.
%Y A320656 counts factorizations into squarefree semiprimes.
%Y The following count vertex-degree partitions and give their Heinz numbers:
%Y - A058696 counts partitions of 2n (A300061).
%Y - A000070 counts non-multigraphical partitions of 2n (A339620 [this sequence]).
%Y - A209816 counts multigraphical partitions (A320924).
%Y - A147878 counts connected multigraphical partitions (A320925).
%Y - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y - A339656 counts loop-graphical partitions (A339658).
%Y - A339617 counts non-graphical partitions of 2n (A339618).
%Y - A000569 counts graphical partitions (A320922).
%Y The following count partitions of even length and give their Heinz numbers:
%Y - A027187 has no additional conditions (A028260).
%Y - A096373 cannot be partitioned into strict pairs (A320891).
%Y - A338914 can be partitioned into strict pairs (A320911).
%Y - A338915 cannot be partitioned into distinct pairs (A320892).
%Y - A338916 can be partitioned into distinct pairs (A320912).
%Y - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y - A339560 can be partitioned into distinct strict pairs (A339561).
%Y Cf. A001055, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.
%K nonn
%O 1,1
%A _Gus Wiseman_, Dec 18 2020
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