OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
A 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.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(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 > 1 together with their prime indices begins:
3: {2} 70: {1,3,4} 192: {1,1,1,1,1,1,2}
4: {1,1} 75: {2,3,3} 196: {1,1,4,4}
9: {2,2} 81: {2,2,2,2} 198: {1,2,2,5}
10: {1,3} 84: {1,1,2,4} 208: {1,1,1,1,6}
12: {1,1,2} 88: {1,1,1,5} 210: {1,2,3,4}
16: {1,1,1,1} 90: {1,2,2,3} 220: {1,1,3,5}
25: {3,3} 100: {1,1,3,3} 225: {2,2,3,3}
27: {2,2,2} 108: {1,1,2,2,2} 243: {2,2,2,2,2}
28: {1,1,4} 112: {1,1,1,1,4} 250: {1,3,3,3}
30: {1,2,3} 120: {1,1,1,2,3} 252: {1,1,2,2,4}
36: {1,1,2,2} 144: {1,1,1,1,2,2} 256: {1,1,1,1,1,1,1,1}
40: {1,1,1,3} 147: {2,4,4} 264: {1,1,1,2,5}
48: {1,1,1,1,2} 160: {1,1,1,1,1,3} 270: {1,2,2,2,3}
63: {2,2,4} 175: {3,3,4} 280: {1,1,1,3,4}
64: {1,1,1,1,1,1} 189: {2,2,2,4} 300: {1,1,2,3,3}
For example, the four loop-graphs with degrees y = (3,1,1,1) are:
{{1,1},{1,2},{3,4}}
{{1,1},{1,3},{2,4}}
{{1,1},{1,4},{2,3}}
{{1,2},{1,3},{1,4}},
so the Heinz number 40 is in the sequence. On the other hand, the three loop-multigraphs with degrees y = (4,4) are
{{1,1},{1,1},{2,2},{2,2}}
{{1,1},{1,2},{1,2},{2,2}}
{{1,2},{1,2},{1,2},{1,2}},
but none of these is a loop-graph, so the Heinz number 49 is not 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[25], Select[mpsbin[nrmptn[#]], UnsameQ@@#&]!={}&]
CROSSREFS
A339656 counts these partitions.
A101048 counts partitions into semiprimes.
A320655 counts factorizations into 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