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a(n) = A102640(n) + A102641(n) - 1.
4

%I #12 Nov 27 2021 11:06:31

%S 2,3,5,5,4,5,3,4,4,5,3,7,3,6,3,3,4,6,3,6,4,7,3,6,3,5,3,5,4,7,3,5,4,5,

%T 5,7,3,6,5,5,3,6,3,6,5,5,3,7,3,5,4,5,3,6,7,3,6,3,3,4,3,5,4,3,4,6,3,5,

%U 5,6,3,7,3,5,7,5,5,6,3,5,6,5,3,4,3,5,3,6,3,5,3,5,4,5,5,5,4,6,4,7,3,6,3,4,7

%N a(n) = A102640(n) + A102641(n) - 1.

%C A006530(2^n)=2 is a local minimum. Actual sequence displays the "width of valley" between the two nearest peaks of largest prime divisors. At the bottom of valley lies the number 2, the minimum.

%e n=12: 2^10=4096. The greatest prime divisors of numbers around 4096 [both downward and upward] are as follows: {31, 4093, 89, 13, 2, 241, 683, 4099, 41}. The length of relevant sequence, i.e., between peaks 4093 and 4099 is 7, thus a(12)=7.

%t With[{nn = 12, lim = 105}, Map[Total@ # - 1 &, Transpose@ {Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Array[FactorInteger[#][[-1, 1]] &, nn, k]][2^n], {n, lim}], Table[Function[k, 1 + LengthWhile[#, # > 0 &] &@ Differences@ Table[FactorInteger[m][[-1, 1]], {m, k, k - nn, -1}]][2^n], {n, lim}]}]] (* _Michael De Vlieger_, Jul 30 2017 *)

%Y Cf. A006530, A102640, A102641, A102643, A102644.

%K nonn

%O 1,1

%A _Labos Elemer_, Jan 21 2005