A353844 Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Author

Gus Wiseman, May 26 2022

Keywords

• nonn

Comments

The run-sums transformation is described by Kimberling at A237685 and A237750.

The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.

Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.

Examples

			The terms together with their prime indices begin:
      1: {}            25: {3,3}           64: {1,1,1,1,1,1}
      2: {1}           27: {2,2,2}         67: {19}
      3: {2}           29: {10}            71: {20}
      4: {1,1}         31: {11}            73: {21}
      5: {3}           32: {1,1,1,1,1}     79: {22}
      7: {4}           37: {12}            81: {2,2,2,2}
      8: {1,1,1}       40: {1,1,1,3}       83: {23}
      9: {2,2}         41: {13}            84: {1,1,2,4}
     11: {5}           43: {14}            89: {24}
     12: {1,1,2}       47: {15}            97: {25}
     13: {6}           49: {4,4}          101: {26}
     16: {1,1,1,1}     53: {16}           103: {27}
     17: {7}           59: {17}           107: {28}
     19: {8}           61: {18}           109: {29}
     23: {9}           63: {2,2,4}        112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.
		

Crossrefs

This sequence is a subset of A300273, counted by A275870.

The version for compositions is A353857, counted by A353847.

A001222 counts prime factors, distinct A001221.

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

A124010 gives prime signature, sorted A118914.

A304442 counts partitions with all equal run-sums.

A353851 counts compositions with all equal run-sums, ranked by A353848.

A325268 counts partitions by omicron, rank statistic A304465.

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

A353833 ranks partitions with all equal run-sums, nonprime A353834.

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

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

A353840-A353846 pertain to partition run-sum trajectory.

A353853-A353859 pertain to composition run-sum trajectory.

A353866 ranks rucksack partitions, counted by A353864.

Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.

Programs

Mathematica

ope[n_]:=Times@@Prime/@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]*k];
Select[Range[100],#==1||PrimeQ[NestWhile[ope,#,!SquareFreeQ[#]&]]&]