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A339618 Heinz numbers of non-graphical integer partitions of even numbers.

%I A339618 #12 Feb 16 2025 08:34:01
%S A339618 3,7,9,10,13,19,21,22,25,28,29,30,34,37,39,43,46,49,52,53,55,57,61,62,
%T A339618 63,66,70,71,75,76,79,82,84,85,87,88,89,91,94,100,101,102,107,111,113,
%U A339618 115,116,117,118,121,129,130,131,133,134,136,138,139,146,147
%N A339618 Heinz numbers of non-graphical integer partitions of even numbers.
%C A339618 An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.
%C A339618 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 A339618 The following are equivalent characteristics for any positive integer n:
%C A339618 (1) the multiset of prime indices of n can be partitioned into distinct strict pairs (a set of edges);
%C A339618 (2) n can be factored into distinct squarefree semiprimes;
%C A339618 (3) the unordered prime signature of n is graphical.
%H A339618 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339618 Equals A300061 \ A320922.
%F A339618 For all n, A181821(a(n)) and A304660(a(n)) belong to A320894.
%e A339618 The sequence of terms together with their prime indices begins:
%e A339618       3: {2}         43: {14}        79: {22}
%e A339618       7: {4}         46: {1,9}       82: {1,13}
%e A339618       9: {2,2}       49: {4,4}       84: {1,1,2,4}
%e A339618      10: {1,3}       52: {1,1,6}     85: {3,7}
%e A339618      13: {6}         53: {16}        87: {2,10}
%e A339618      19: {8}         55: {3,5}       88: {1,1,1,5}
%e A339618      21: {2,4}       57: {2,8}       89: {24}
%e A339618      22: {1,5}       61: {18}        91: {4,6}
%e A339618      25: {3,3}       62: {1,11}      94: {1,15}
%e A339618      28: {1,1,4}     63: {2,2,4}    100: {1,1,3,3}
%e A339618      29: {10}        66: {1,2,5}    101: {26}
%e A339618      30: {1,2,3}     70: {1,3,4}    102: {1,2,7}
%e A339618      34: {1,7}       71: {20}       107: {28}
%e A339618      37: {12}        75: {2,3,3}    111: {2,12}
%e A339618      39: {2,6}       76: {1,1,8}    113: {30}
%e A339618 For example, there are three possible multigraphs with degrees (1,1,3,3):
%e A339618   {{1,2},{1,2},{1,2},{3,4}}
%e A339618   {{1,2},{1,2},{1,3},{2,4}}
%e A339618   {{1,2},{1,2},{1,4},{2,3}}.
%e A339618 Since none of these is a graph, the Heinz number 100 belongs to the sequence.
%t A339618 strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
%t A339618 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A339618 Select[Range[100],EvenQ[Length[nrmptn[#]]]&&strs[Times@@Prime/@nrmptn[#]]=={}&]
%Y A339618 A181819 applied to A320894 gives this sequence.
%Y A339618 A300061 is a superset.
%Y A339618 A339617 counts these partitions.
%Y A339618 A320922 ranks the complement, counted by A000569.
%Y A339618 A006881 lists squarefree semiprimes.
%Y A339618 A320656 counts factorizations into squarefree semiprimes.
%Y A339618 A339659 counts graphical partitions of 2n into k parts.
%Y A339618 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339618 - A058696 counts partitions of 2n (A300061).
%Y A339618 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339618 - A209816 counts multigraphical partitions (A320924).
%Y A339618 - A339655 counts non-loop-graphical partitions of 2n (A339657).
%Y A339618 - A339656 counts loop-graphical partitions (A339658).
%Y A339618 - A339617 counts non-graphical partitions of 2n (A339618 [this sequence]).
%Y A339618 - A000569 counts graphical partitions (A320922).
%Y A339618 The following count partitions of even length and give their Heinz numbers:
%Y A339618 - A027187 has no additional conditions (A028260).
%Y A339618 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339618 - A338914 can be partitioned into strict pairs (A320911).
%Y A339618 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339618 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339618 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339618 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339618 Cf. A001358, A007717, A050326, A320923, A338899.
%K A339618 nonn
%O A339618 1,1
%A A339618 _Gus Wiseman_, Dec 18 2020