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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.
EXAMPLE
Triangle begins:
1
2
3
4 3
5
6
7
8 5
9 7
10
11
12 9 7
Row 87780 is the following trajectory (left column), with prime indices shown on the right:
87780: {1,1,2,3,4,5,8}
65835: {2,2,3,4,5,8}
51205: {3,4,4,5,8}
19855: {3,5,8,8}
2915: {3,5,16}
MATHEMATICA
Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#], {p_, k_}:>PrimePi[p]*k]&, n, Not@*SquareFreeQ], {n, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 25 2022
STATUS
approved