A353841 Length of the trajectory of the partition run-sum transformation of n, using Heinz numbers; a(1) = 0.

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1

Offset

1,4

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.

Starting with n, this is one plus the number of times one must apply A353832 to reach a squarefree number.

Also Kimberling's depth statistic (defined in A237685 and A237750) plus one.

Links

Table of n, a(n) for n=1..87.

Example

The trajectory for a(1080) = 4 is the following, with prime indices shown on the right:
  1080: {1,1,1,2,2,2,3}
   325: {3,3,6}
   169: {6,6}
    37: {12}
The trajectory for a(87780) = 5 is the following, 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}
The trajectory for a(39960) = 5 is the following, with prime indices shown on the right:
  39960: {1,1,1,2,2,2,3,12}
  12025: {3,3,6,12}
   6253: {6,6,12}
   1369: {12,12}
     89: {24}

Mathematica

Table[If[n==1, 0, Length[NestWhileList[Times@@Prime/@Cases[If[#==1, {}, FactorInteger[#]], {p_, k_}:>PrimePi[p]*k]&, n, !SquareFreeQ[#]&]]], {n, 100}]

Crossrefs

Positions of 1's are A005117.

The version for run-lengths instead of sums is A182850 or A323014.

Positions of first appearances are A353743.

These are the row-lengths of A353840.

Other sequences pertaining to this trajectory are A353842-A353845.

Counting partitions by this statistic gives A353846.

The version for compositions is A353854, run-lengths of A353853.

A001222 counts prime factors, distinct A001221.

A005811 counts runs in binary expansion.

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

A300273 ranks collapsible partitions, counted by A275870.

A318928 gives runs-resistance of binary expansion.

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.

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

A353866 ranks rucksack partitions, counted by A353864.

Cf. A071625, A073093, A181819, A182857, A325239, A325277, A325278, A353834, A353847, A353865, A353867.

Sequence in context: A316975 A043277 A265991 * A322871 A322870 A170979

Adjacent sequences: A353838 A353839 A353840 * A353842 A353843 A353844

Keyword

nonn

Author

Gus Wiseman, May 25 2022

Status

approved