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

Author

Gus Wiseman, Jun 11 2022

Keywords

• nonn

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).

Examples

			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}
		

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.

Programs

Mathematica

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

Formula

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

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