
COMMENTS

The sequence is infinite since, for example, for any n >= 1 the symmetric representation of sigma(3^n) consists of n + 1 parts of width 1. However, it is not increasing since a(11) = 59049 = 3^10 and a(12) = 29095 = 5 * 11 * 23^2. Also a(13) <= 531441 = 3^12.
This sequence is a subsequence of A174905; its subsequences a(n) for odd/even n are subsequences of A241010/A241008, respectively. Some evenindexed elements of this sequence are members of A239663, e.g., a(2), a(4), a(6), a(8) and a(12), but not a(10) = 6875.
The central pair of parts in the symmetric representation of sigma(a(2)), sigma(a(4)) and sigma(a(8)) meets at the diagonal (see A298856).


MATHEMATICA

(* Function path[] is defined in A237270 *)
segmentsSR[pathN0_, pathN1_] := SplitBy[Map[Min, Drop[Drop[pathN0, 1], 1]  pathN1], #==0&]
regions[pathN0_ , pathN1_] := Select[Map[Apply[Plus, #]&, segmentsSR[pathN0, pathN1]], #!=0&]
width1Q[pathN0_, pathN1_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[pathN0, 1], 1]  pathN1, 1]]]
(* parameter seq is the list of elements of the sequence in interval 1..m1 already computed with an entry of 0 representing an element not yet found *)
a318843[m_, n_, seq_] := Module[{list=Join[seq, Table[0, 10]], path1=path[m1], path0, k, a, r, w}, For[k=m, k<=n, k++, path0=path[k]; a=regions[path0, path1]; r=Length[a]; w=width1Q[path0, path1]; If[w && list[[r]]==0, list[[r]]=k]; path1=path0]; list]
a318843[2, 60000, {1}] (* data  actually computed in steps *)
