
COMMENTS

It can be shown by an application of Mihailescu's theorem that a(12) does not exist, since then there would be two 3smooth numbers close together (it suffices to check up to 2*3^3).
If a(9) exists, it is greater than 10^30.  Don Reble (djr(AT)nk.ca), Mar 02 2003
If a(9) exists, it is greater than 10^3000.  Charles R Greathouse IV, Apr 22 2009
Eggleton and MacDougall prove that no terms exist beyond a(9) and conjecture that a(9) does not exist.  Jason Kimberley, Jul 08 2017


MATHEMATICA

Function[s, Function[t, Reverse@ FoldList[If[#2 > #1, #1, #2] &, Reverse[#]] &@ Map[t[[First@ FirstPosition[t[[All, 1]], k_ /; k == #] ]] &, Range[0, Max@ t[[All, 1]] ] ][[All, 1]] ]@ Join[{{First@ s, 0}, {#[[1, 1, 1]], 1}}, Rest@ Map[{#[[1, 1]], Length@ # + 1} &, #, {1}]] &@ SplitBy[Partition[Select[#, Last@ # == 1 &][[All, 1]], 2, 1], Differences] &@ Map[{First@ #, First@ Differences@ #} &, Partition[s, 2, 1]]]@ Select[Range[10^5], PrimeNu[#] == 2 &] (* Michael De Vlieger, Jul 17 2017 *)
