A353743 Least number with run-sum trajectory of length k; a(0) = 1.

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset

0,2

Comments

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

Links

Table of n, a(n) for n=0..15.

Formula

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

Example

The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

Mathematica

Join[{1, 2}, Table[2*Product[Prime[2^k], {k, 0, n}], {n, 0, 6}]]

Crossrefs

The ordered version is A072639, for run-lengths A333629.

The version for run-lengths is A325278, firsts in A182850 or A323014.

The run-sum trajectory is the iteration of A353832.

The first length-k row of A353840 has index a(k).

Other sequences pertaining to this trajectory are A353841-A353846.

A001222 counts prime factors, distinct A001221.

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

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.

A353866 ranks rucksack partitions, counted by A353864.

Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Sequence in context: A001206 A144295 A119489 * A217757 A053631 A319634

Adjacent sequences: A353740 A353741 A353742 * A353744 A353745 A353746

Keyword

nonn

Author

Gus Wiseman, Jun 11 2022

Status

approved