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

7, 13, 19, 21, 22, 29, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 154, 155, 156, 159, 163, 165, 166, 169, 171

Offset: 1

Author

Gus Wiseman, Dec 18 2020

Keywords

• nonn

Comments

Equals the image of A181819 applied to the set of terms of A320892.

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.

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Graphical partition.

Crossrefs

A320892 has these prime shadows (see A181819).

A321728 is conjectured to be the version for half-loops {x} instead of loops {x,x}.

A339655 counts these partitions.

A339658 ranks the complement, counted by A339656.

A001358 lists semiprimes, with odd and even terms A046315 and A100484.

A006881 lists squarefree semiprimes, with odd and even terms A046388 and A100484.

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:

- A058696 counts partitions of 2n (A300061).

- A000070 counts non-multigraphical partitions of 2n (A339620).

- A209816 counts multigraphical partitions (A320924).

- A339655 counts non-loop-graphical partitions of 2n (A339657 [this sequence]).

- A339656 counts loop-graphical partitions (A339658).

- A339617 counts non-graphical partitions of 2n (A339618).

- A000569 counts graphical partitions (A320922).

The following count partitions of even length and give their Heinz numbers:

- A027187 has no additional conditions (A028260).

- A096373 cannot be partitioned into strict pairs (A320891).

- A338914 can be partitioned into strict pairs (A320911).

- A338915 cannot be partitioned into distinct pairs (A320892).

- A338916 can be partitioned into distinct pairs (A320912).

- A339559 cannot be partitioned into distinct strict pairs (A320894).

- A339560 can be partitioned into distinct strict pairs (A339561).

Cf. A001055, A001221, A001222, A007717, A056239, A112798, A320732, A338898, A338912, A338913, A339742, A339839.

Programs

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@@#&]=={}&]

Formula

A300061 = A339657 \/ A339658.