OFFSET
1,1
COMMENTS
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.
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.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.
LINKS
Eric Weisstein's World of Mathematics, Graphical partition.
EXAMPLE
The sequence of terms together with their prime indices begins:
7: {4} 57: {2,8} 107: {28}
13: {6} 61: {18} 111: {2,12}
19: {8} 62: {1,11} 113: {30}
21: {2,4} 66: {1,2,5} 115: {3,9}
22: {1,5} 71: {20} 116: {1,1,10}
29: {10} 76: {1,1,8} 117: {2,2,6}
34: {1,7} 79: {22} 118: {1,17}
37: {12} 82: {1,13} 121: {5,5}
39: {2,6} 85: {3,7} 129: {2,14}
43: {14} 87: {2,10} 130: {1,3,6}
46: {1,9} 89: {24} 131: {32}
49: {4,4} 91: {4,6} 133: {4,8}
52: {1,1,6} 94: {1,15} 134: {1,19}
53: {16} 101: {26} 136: {1,1,1,7}
55: {3,5} 102: {1,2,7} 138: {1,2,9}
For example, the three loop-multigraphs with degrees y = (5,2,1) are:
{{1,1},{1,1},{1,2},{2,3}}
{{1,1},{1,1},{1,3},{2,2}}
{{1,1},{1,2},{1,2},{1,3}},
but since none of these is a loop-graph (they have multiple edges), the Heinz number 66 is in the sequence.
MATHEMATICA
spsbin[{}]:={{}}; spsbin[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@spsbin[Complement[set, s]]]/@Cases[Subsets[set], {i, _}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[50], EvenQ[Length[nrmptn[#]]]&&Select[mpsbin[nrmptn[#]], UnsameQ@@#&]=={}&]
CROSSREFS
A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.
A339655 counts these partitions.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339844 counts loop-graphical partitions by length.
factorizations of n into distinct primes or squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
The following count partitions of even length and give their Heinz numbers:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 18 2020
STATUS
approved