A390569 Odd Heinz numbers of multisets that are not the first sums of any nonempty multiset.

25, 75, 121, 125, 147, 175, 225, 231, 275, 289, 325, 363, 375, 425, 429, 441, 475, 507, 525, 529, 561, 575, 605, 625, 663, 675, 693, 725, 741, 775, 825, 845, 847, 867, 875, 897, 925, 961, 969, 975, 1025, 1029, 1075, 1083, 1089, 1105, 1125, 1173, 1175, 1225

Offset

1,1

Comments

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

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.

Links

Table of n, a(n) for n=1..50.

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.The terms together with their prime indices begin:
    25: {3,3}
    75: {2,3,3}
   121: {5,5}
   125: {3,3,3}
   147: {2,4,4}
   175: {3,3,4}
   225: {2,2,3,3}
   231: {2,4,5}
   275: {3,3,5}
   289: {7,7}
   325: {3,3,6}
   363: {2,5,5}
   375: {2,3,3,3}
   425: {3,3,7}
   429: {2,5,6}
   441: {2,2,4,4}
   475: {3,3,8}
   507: {2,6,6}
   525: {2,3,3,4}

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[1, 1000, 2], With[{m=prix[#]}, Select[Table[uncha[m, i], {i, 1, Max[m]}], LessEqual@@#&]=={}]&]

Crossrefs

These partitions are counted by A390431.

This is the odd case of A390445 (complement of A390449), counted by A390447.

The complement allowing evens is A390448 counted by A390446.

A000041 counts integer partitions, strict A000009.

A112798 lists prime indices, sum A056239, multiplicities A124010.

A342527 counts compositions with all equal first sums.

A390307 gives first sums of prime indices.

A390567 counts compositions with all distinct first sums.

Cf. A001222, A049988, A055396, A215366, A325325, A325328, A389811, A389919, A390362, A390429, A390444.

Sequence in context: A044544 A045180 A270385 * A053742 A123296 A118610

Adjacent sequences: A390566 A390567 A390568 * A390570 A390571 A390572

Keyword

nonn

Author

Gus Wiseman, Nov 19 2025

Status

approved