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A339657 Heinz numbers of non-loop-graphical partitions of even numbers.

%I A339657 #15 Feb 16 2025 08:34:01
%S A339657 7,13,19,21,22,29,34,37,39,43,46,49,52,53,55,57,61,62,66,71,76,79,82,
%T A339657 85,87,89,91,94,101,102,107,111,113,115,116,117,118,121,129,130,131,
%U A339657 133,134,136,138,139,146,148,151,154,155,156,159,163,165,166,169,171
%N A339657 Heinz numbers of non-loop-graphical partitions of even numbers.
%C A339657 Equals the image of A181819 applied to the set of terms of A320892.
%C A339657 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.
%C A339657 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. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.
%C A339657 The following are equivalent characteristics for any positive integer n:
%C A339657 (1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
%C A339657 (2) n can be factored into distinct semiprimes;
%C A339657 (3) the prime signature of n is loop-graphical.
%H A339657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphicalPartition.html">Graphical partition.</a>
%F A339657 A300061 = A339657 \/ A339658.
%e A339657 The sequence of terms together with their prime indices begins:
%e A339657       7: {4}         57: {2,8}      107: {28}
%e A339657      13: {6}         61: {18}       111: {2,12}
%e A339657      19: {8}         62: {1,11}     113: {30}
%e A339657      21: {2,4}       66: {1,2,5}    115: {3,9}
%e A339657      22: {1,5}       71: {20}       116: {1,1,10}
%e A339657      29: {10}        76: {1,1,8}    117: {2,2,6}
%e A339657      34: {1,7}       79: {22}       118: {1,17}
%e A339657      37: {12}        82: {1,13}     121: {5,5}
%e A339657      39: {2,6}       85: {3,7}      129: {2,14}
%e A339657      43: {14}        87: {2,10}     130: {1,3,6}
%e A339657      46: {1,9}       89: {24}       131: {32}
%e A339657      49: {4,4}       91: {4,6}      133: {4,8}
%e A339657      52: {1,1,6}     94: {1,15}     134: {1,19}
%e A339657      53: {16}       101: {26}       136: {1,1,1,7}
%e A339657      55: {3,5}      102: {1,2,7}    138: {1,2,9}
%e A339657 For example, the three loop-multigraphs with degrees y = (5,2,1) are:
%e A339657   {{1,1},{1,1},{1,2},{2,3}}
%e A339657   {{1,1},{1,1},{1,3},{2,2}}
%e A339657   {{1,1},{1,2},{1,2},{1,3}},
%e A339657 but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.
%t A339657 spsbin[{}]:={{}};spsbin[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
%t A339657 mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
%t A339657 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A339657 Select[Range[50],EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]],UnsameQ@@#&]=={}&]
%Y A339657 A320892 has these prime shadows (see A181819).
%Y A339657 A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
%Y A339657 A339655 counts these partitions.
%Y A339657 A339658 ranks the complement, counted by A339656.
%Y A339657 A001358 lists semiprimes, with odd and even terms A046315 and A100484.
%Y A339657 A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.
%Y A339657 A101048 counts partitions into semiprimes.
%Y A339657 A320655 counts factorizations into semiprimes.
%Y A339657 A320656 counts factorizations into squarefree semiprimes.
%Y A339657 A339844 counts loop-graphical partitions by length.
%Y A339657 factorizations of n into distinct primes or squarefree semiprimes.
%Y A339657 The following count vertex-degree partitions and give their Heinz numbers:
%Y A339657 - A058696 counts partitions of 2n (A300061).
%Y A339657 - A000070 counts non-multigraphical partitions of 2n (A339620).
%Y A339657 - A209816 counts multigraphical partitions (A320924).
%Y A339657 - A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).
%Y A339657 - A339656 counts loop-graphical partitions (A339658).
%Y A339657 - A339617 counts non-graphical partitions of 2n (A339618).
%Y A339657 - A000569 counts graphical partitions (A320922).
%Y A339657 The following count partitions of even length and give their Heinz numbers:
%Y A339657 - A027187 has no additional conditions (A028260).
%Y A339657 - A096373 cannot be partitioned into strict pairs (A320891).
%Y A339657 - A338914 can be partitioned into strict pairs (A320911).
%Y A339657 - A338915 cannot be partitioned into distinct pairs (A320892).
%Y A339657 - A338916 can be partitioned into distinct pairs (A320912).
%Y A339657 - A339559 cannot be partitioned into distinct strict pairs (A320894).
%Y A339657 - A339560 can be partitioned into distinct strict pairs (A339561).
%Y A339657 Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.
%K A339657 nonn
%O A339657 1,1
%A A339657 _Gus Wiseman_, Dec 18 2020