A390445 Heinz numbers of integer partitions that are not the first sums of any nonempty multiset.

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 25, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 121

Offset

1,1

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.

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Links

Table of n, a(n) for n=1..63.

Example

The first sums of (1,1,1,2) are (2,2,3), which is the prime indices of 45, so 45 is not in the sequence.
The first sums of (3,3,3) are (6,6), which is the prime indices of 169, so 169 is not in the sequence.

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, 1, Length[tar]}]; cur);
Select[Range[2, 100], With[{m=prix[#]}, Select[Table[uncha[m, i], {i, 1, Max[m]}], LessEqual@@#&]=={}]&]

Crossrefs

These are numbers whose prime indices are not a row of A390307.

Positions of 0 in A390444 with alternating zeros, except 1.

These partitions are counted by A390447.

The complement is A390448, counted by A390446.

Complement of A390449.

The odd case is A390569, counted by A390431.

A000041 counts integer partitions, strict A000009.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A390307 gives first sums of prime indices.

A390429 counts distinct first sums of partitions.

Cf. A001222, A004709, A055396, A215366, A243055, A325325, A325328, A355536, A368684, A389599, A390362.

Sequence in context: A337718 A246410 A396013 * A195169 A338922 A055964

Adjacent sequences: A390442 A390443 A390444 * A390446 A390447 A390448

Keyword

nonn

Author

Gus Wiseman, Nov 13 2025

Status

approved