A363531 Heinz numbers of integer partitions such that 3*(sum) = (reverse-weighted sum).

1, 32, 144, 216, 243, 672, 1008, 1350, 2176, 2250, 2520, 2673, 3125, 3969, 4160, 4200, 5940, 6240, 6615, 7344, 7424, 7744, 8262, 9261, 9800, 9900, 10400, 11616, 12250, 12312, 12375, 13104, 13720, 14720, 14742, 16767, 16807, 17150, 19360, 21840, 22080, 23100

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. The reverse-weighted sum is the weighted sum of the reverse, also the sum of partial sums. For example, the weighted sum of (4,2,2,1) is 1*4 + 2*2 + 3*2 + 4*1 = 18 and the reverse-weighted sum is 4*4 + 3*2 + 2*2 + 1*1 = 27.

Links

Table of n, a(n) for n=1..42.

Formula

A056239(a(n)) = A318283(a(n))/3.

Example

The terms together with their prime indices begin:
      1: {}
     32: {1,1,1,1,1}
    144: {1,1,1,1,2,2}
    216: {1,1,1,2,2,2}
    243: {2,2,2,2,2}
    672: {1,1,1,1,1,2,4}
   1008: {1,1,1,1,2,2,4}
   1350: {1,2,2,2,3,3}
   2176: {1,1,1,1,1,1,1,7}
   2250: {1,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   2673: {2,2,2,2,2,5}
   3125: {3,3,3,3,3}
   3969: {2,2,2,2,4,4}
   4160: {1,1,1,1,1,1,3,6}

Mathematica

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

Crossrefs

These partitions are counted by A363526.

The non-reverse version is A363530, counted by A363527.

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: A159763 A100164 A048191 * A224041 A188839 A188702

Adjacent sequences: A363528 A363529 A363530 * A363532 A363533 A363534

Keyword

nonn

Author

Gus Wiseman, Jun 12 2023

Status

approved