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

%I A353743 #10 Jun 12 2022 22:52:18
%S A353743 1,2,4,12,84,1596,84588,11081028,3446199708,2477817590052,
%T A353743 4011586678294188,14726534696017964148,120183249654202605411828,
%U A353743 2146833388573021140471483564,83453854313999050793547980583372,7011542477899258250521520684673165324
%N A353743 Least number with run-sum trajectory of length k; a(0) = 1.
%C A353743 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 A353743 a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).
%F A353743 a(n > 0) = 2 * A325782(n).
%e A353743 The terms together with their prime indices begin:
%e A353743       1: {}
%e A353743       2: {1}
%e A353743       4: {1,1}
%e A353743      12: {1,1,2}
%e A353743      84: {1,1,2,4}
%e A353743    1596: {1,1,2,4,8}
%e A353743   84588: {1,1,2,4,8,16}
%t A353743 Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]
%Y A353743 The ordered version is A072639, for run-lengths A333629.
%Y A353743 The version for run-lengths is A325278, firsts in A182850 or A323014.
%Y A353743 The run-sum trajectory is the iteration of A353832.
%Y A353743 The first length-k row of A353840 has index a(k).
%Y A353743 Other sequences pertaining to this trajectory are A353841-A353846.
%Y A353743 A001222 counts prime factors, distinct A001221.
%Y A353743 A056239 adds up prime indices, row sums of A112798 and A296150.
%Y A353743 A300273 ranks collapsible partitions, counted by A275870.
%Y A353743 A353833 ranks partitions with all equal run-sums, counted by A304442.
%Y A353743 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A353743 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353743 A353866 ranks rucksack partitions, counted by A353864.
%Y A353743 Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.
%K A353743 nonn
%O A353743 0,2
%A A353743 _Gus Wiseman_, Jun 11 2022