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.
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 12 2023
STATUS
approved