A339620 Heinz numbers of non-multigraphical partitions of even numbers.

3, 7, 10, 13, 19, 21, 22, 28, 29, 34, 37, 39, 43, 46, 52, 53, 55, 57, 61, 62, 66, 71, 76, 79, 82, 85, 87, 88, 89, 91, 94, 101, 102, 107, 111, 113, 115, 116, 117, 118, 129, 130, 131, 133, 134, 136, 138, 139, 146, 148, 151, 155, 156, 159, 163, 166, 171, 172, 173

Offset

1,1

Comments

An integer partition is non-multigraphical if it does not comprise the multiset of vertex-degrees of any multigraph (multiset of non-loop edges). Multigraphical partitions are counted by A209816, non-multigraphical partitions by A000070.

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.

The following are equivalent characteristics for any positive integer n:

(1) the multiset of prime indices of n can be partitioned into strict pairs (a multiset of edges);

(2) n can be factored into squarefree semiprimes;

(3) the unordered prime signature of n is multigraphical.

Links

Table of n, a(n) for n=1..59.

Eric Weisstein's World of Mathematics, Graphical partition.

Formula

Equals A300061 \ A320924.

For all n, both A181821(a(n)) and A304660(a(n)) belong to A320891.

Example

The sequence of terms together with their prime indices begins:
      3: {2}         53: {16}          94: {1,15}
      7: {4}         55: {3,5}        101: {26}
     10: {1,3}       57: {2,8}        102: {1,2,7}
     13: {6}         61: {18}         107: {28}
     19: {8}         62: {1,11}       111: {2,12}
     21: {2,4}       66: {1,2,5}      113: {30}
     22: {1,5}       71: {20}         115: {3,9}
     28: {1,1,4}     76: {1,1,8}      116: {1,1,10}
     29: {10}        79: {22}         117: {2,2,6}
     34: {1,7}       82: {1,13}       118: {1,17}
     37: {12}        85: {3,7}        129: {2,14}
     39: {2,6}       87: {2,10}       130: {1,3,6}
     43: {14}        88: {1,1,1,5}    131: {32}
     46: {1,9}       89: {24}         133: {4,8}
     52: {1,1,6}     91: {4,6}        134: {1,19}
For example, a complete lists of all loop-multigraphs with degrees (5,2,1) is:
  {{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 multigraph (they have loops), the Heinz number 66 belongs to the sequence.

Mathematica

prpts[m_]:=If[Length[m]==0, {{}}, Join@@Table[Prepend[#, ipr]&/@prpts[Fold[DeleteCases[#1, #2, {1}, 1]&, m, ipr]], {ipr, Select[Subsets[Union[m], {2}], MemberQ[#, m[[1]]]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], EvenQ[Length[nrmptn[#]]]&&prpts[nrmptn[#]]=={}&]

Crossrefs

A000070 counts these partitions.

A300061 is a superset.

A320891 has image under A181819 equal to this set of terms.

A001358 lists semiprimes, with squarefree case A006881.

A002100 counts partitions into squarefree semiprimes.

A320656 counts factorizations into 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 [this sequence]).

- A209816 counts multigraphical partitions (A320924).

- A147878 counts connected multigraphical partitions (A320925).

- A339655 counts non-loop-graphical partitions of 2n (A339657).

- 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, A005117, A007717, A030229, A050320, A056239, A112798, A320655, A338899, A339113, A339661.

Sequence in context: A285359 A255607 A310185 * A319241 A310186 A363022

Adjacent sequences: A339617 A339618 A339619 * A339621 A339622 A339623

Keyword

nonn

Author

Gus Wiseman, Dec 18 2020

Status

approved