A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

Offset: 1

Author

Gus Wiseman, May 25 2022

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 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.

Examples

			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}
		

Crossrefs

The version for run-lengths instead of sums is A325239 or A325277.

This is the iteration of A353832, with composition version A353847.

Row-lengths are A353841, counted by A353846.

Final terms are A353842.

Counting rows by final omega gives A353843.

Rows ending in a prime number are A353844, counted by A353845.

These sequences for compositions are A353853-A353859.

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798 and A296150.

A124010 gives prime signature, sorted A118914.

A182850 or A323014 gives frequency depth.

A300273 ranks collapsible partitions, counted by A275870.

A353833 ranks partitions with all equal run-sums, counted by A304442.

A353835 counts distinct run-sums of prime indices, weak A353861.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353862 gives greatest run-sum of prime indices, least A353931.

Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Programs

Mathematica

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]