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%I #33 Dec 31 2023 00:17:01
%S 1,3,9,21,81,147,729,903,3025,6875,59049,29095,531441,171875,366025,
%T 643885,43046721,3511475
%N a(n) is the smallest number k such that the symmetric representation of sigma(k) consists of n parts of width 1.
%C 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.
%C This sequence is a subsequence of A174905; its subsequences a(n) for odd/even n are subsequences of A241010/A241008, respectively. Some even-indexed 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.
%C 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).
%C From _Hartmut F. W. Hoft_, Oct 04 2021: (Start)
%C An upper bound to the sequence is a(n) <= 3^(n-1), n >= 1, (see A348171).
%C For p = 1,2,3,5,7,11,13,17, a(p) = 3^(p-1) and this equality possibly holds for all a(p) with p a prime.
%C Also, 75 * 10^6 < a(19) <= 3^18, a(20) = 15391255, a(21) = 44289025 and a(n) > 75 * 10^6 for n > 21.
%C a(13)-a(18) computations based on A348171 rather than A237270.
%C The symmetric representation of sigma(3^(p-1)), p prime, consists of p parts and its middle part has area 3^((p-1)/2). (End)
%C a(n) >= A038547(n) with equality for n=1 and primes n since the distinct prime divisors of n can be replaced by primes 3, 5, 7, 11, ... yielding a smaller number k with the same number of odd divisors. However, some parts in the symmetric representation of sigma(k) have width at least 2. - _Hartmut F. W. Hoft_, Dec 11 2023
%e The smallest number k whose symmetric representation of sigma(k) consists of four parts of width one is a(4) = 21. The parts are 11, 5, 5, 11.
%e a(4) = 3*7 has width pattern, A341969, 1010101 while A038547(4) = 3*5 has width pattern 1012101. a(6) = 3 * 7^2 = 147 has width pattern 10101010101 while A038547(6) = 3^2 * 5 = 45 has width pattern 10121212101. - _Hartmut F. W. Hoft_, Dec 11 2023
%t (* Function path[] is defined in A237270 *)
%t segmentsSR[pathN0_, pathN1_] := SplitBy[Map[Min, Drop[Drop[pathN0, 1], -1] - pathN1], #==0&]
%t regions[pathN0_ ,pathN1_] := Select[Map[Apply[Plus, #]&, segmentsSR[pathN0, pathN1]], #!=0&]
%t width1Q[pathN0_, pathN1_] := SubsetQ[{0, 1}, Union[Flatten[Drop[Drop[pathN0, 1], -1] - pathN1, 1]]]
%t (* parameter seq is the list of elements of the sequence in interval 1..m-1 already computed with an entry of 0 representing an element not yet found *)
%t a318843[m_, n_, seq_] := Module[{list=Join[seq, Table[0, 10]], path1=path[m-1], 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]
%t a318843[2,60000,{1}] (* data - actually computed in steps *)
%Y Cf. A174905, A237048, A237270, A237271, A237593, A239663, A239665, A240062, A240542, A241008, A241010, A249351, A298856.
%Y Cf. A249223, A348171.
%Y Cf. A038547, A341969.
%K nonn,more
%O 1,2
%A _Hartmut F. W. Hoft_, Sep 04 2018
%E a(13)-a(18) from _Hartmut F. W. Hoft_, Oct 04 2021
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