A353842 Last part of the trajectory of the partition run-sum transformation of n, using Heinz numbers.

1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 7, 13, 14, 15, 7, 17, 14, 19, 15, 21, 22, 23, 15, 13, 26, 13, 21, 29, 30, 31, 11, 33, 34, 35, 21, 37, 38, 39, 13, 41, 42, 43, 33, 35, 46, 47, 21, 19, 26, 51, 39, 53, 26, 55, 35, 57, 58, 59, 35, 61, 62, 19, 13, 65, 66, 67, 51

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 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

Examples

			The partition run-sum trajectory of 87780 is: 87780 -> 65835 -> 51205 -> 19855 -> 2915, so a(87780) = 2915.
		

Crossrefs

The fixed points and image are A005117.

For run-lengths instead of sums we have A304464/A304465, counted by A325268.

These are the row-ends of A353840.

Other sequences pertaining to partition trajectory are A353841-A353846.

The version for compositions is A353855, run-ends of A353853.

A001222 counts prime factors, distinct A001221.

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

A182850 and A323014 give frequency depth.

A300273 ranks collapsible partitions, counted by A275870.

A353832 represents the operation of taking run-sums of a partition.

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

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

A353866 ranks rucksack partitions, counted by A353864.

Cf. A071625, A073093, A181819, A325239, A325277, A353834, A353838, A353847, A353865, A353867.

Programs

Mathematica

Table[NestWhile[Times@@Prime/@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]*k]&,n,!SquareFreeQ[#]&],{n,100}]