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

Author

Gus Wiseman, Jun 12 2023

Keywords

• nonn

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.

Examples

			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}
		

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.

Programs

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[#]]]&]

Formula

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