A363530 Heinz numbers of integer partitions such that 3*(sum) = (weighted sum).

1, 32, 40, 60, 100, 126, 210, 243, 294, 351, 550, 585, 770, 819, 1210, 1274, 1275, 1287, 1521, 1785, 2002, 2366, 2793, 2805, 2875, 3125, 3315, 4025, 4114, 4335, 4389, 4862, 5187, 6325, 6358, 6422, 6783, 7105, 7475, 7581, 8349, 8398, 9386, 9775, 9867, 10925

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.

The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18.

Links

Table of n, a(n) for n=1..46.

Formula

A056239(a(n)) = A304818(a(n))/3.

Example

The terms together with their prime indices begin:
      1: {}
     32: {1,1,1,1,1}
     40: {1,1,1,3}
     60: {1,1,2,3}
    100: {1,1,3,3}
    126: {1,2,2,4}
    210: {1,2,3,4}
    243: {2,2,2,2,2}
    294: {1,2,4,4}
    351: {2,2,2,6}
    550: {1,3,3,5}
    585: {2,2,3,6}
    770: {1,3,4,5}
    819: {2,2,4,6}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], 3*Total[prix[#]]==Total[Accumulate[Reverse[prix[#]]]]&]

Crossrefs

These partitions are counted by A363527.

The reverse version is A363531, counted by A363526.

A053632 counts compositions by weighted sum.

A055396 gives minimum prime index, maximum A061395.

A112798 lists prime indices, length A001222, sum A056239.

A264034 counts partitions by weighted sum, reverse A358194.

A304818 gives weighted sum of prime indices, row-sums of A359361.

A318283 gives weighted sum of reversed prime indices, row-sums of A358136.

A320387 counts multisets by weighted sum, zero-based A359678.

Cf. A000041, A000720, A001221, A046660, A106529, A118914, A124010, A181819, A215366, A359362, A359755.

Sequence in context: A354422 A217060 A326112 * A240246 A167309 A159007

Adjacent sequences: A363527 A363528 A363529 * A363531 A363532 A363533

Keyword

nonn

Author

Gus Wiseman, Jun 12 2023

Status

approved