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Least number with run-sum trajectory of length k; a(0) = 1.
7

%I #10 Jun 12 2022 22:52:18

%S 1,2,4,12,84,1596,84588,11081028,3446199708,2477817590052,

%T 4011586678294188,14726534696017964148,120183249654202605411828,

%U 2146833388573021140471483564,83453854313999050793547980583372,7011542477899258250521520684673165324

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

%C 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).

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

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

%e The terms together with their prime indices begin:

%e 1: {}

%e 2: {1}

%e 4: {1,1}

%e 12: {1,1,2}

%e 84: {1,1,2,4}

%e 1596: {1,1,2,4,8}

%e 84588: {1,1,2,4,8,16}

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

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

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

%Y The run-sum trajectory is the iteration of A353832.

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

%Y Other sequences pertaining to this trajectory are A353841-A353846.

%Y A001222 counts prime factors, distinct A001221.

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

%Y A300273 ranks collapsible partitions, counted by A275870.

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

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

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

%Y A353866 ranks rucksack partitions, counted by A353864.

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

%K nonn

%O 0,2

%A _Gus Wiseman_, Jun 11 2022