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A251621 Run lengths in A249943.

%I A251621 #60 Jan 23 2022 13:54:36
%S A251621 1,1,1,1,1,1,3,1,2,4,1,1,4,6,2,6,4,2,4,6,6,2,6,4,2,6,4,6,8,4,2,4,2,4,
%T A251621 14,4,6,2,10,2,6,6,4,6,6,2,10,2,4,2,12,12,4,2,4,6,2,10,6,6,6,2,6,4,2,
%U A251621 10,14,4,2,4,14,6,10,2,4,6,8,6,6,4,6,8,4
%N A251621 Run lengths in A249943.
%D A251621 Bradley Klee, Posting to Sequence Fans Mailing List, Dec 07 2014
%D A251621 Vladimir Shevelev, Postings to Sequence Fans Mailing List, Dec 07, 10 and 11, 2014
%H A251621 Reinhard Zumkeller, <a href="/A251621/b251621.txt">Table of n, a(n) for n = 1..1000</a>
%H A251621 David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, <a href="http://arxiv.org/abs/1501.01669">The Yellowstone Permutation</a>, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Sloane/sloane9.html">J. Int. Seq. 18 (2015) 15.6.7</a>.
%F A251621 Connection with prime gaps: conjecturally, for n>=13, we have a(n) = A001223(n-5). - _Vladimir Shevelev_, Dec 07 2014
%F A251621 _Bradley Klee_ noted that this conjecture and his conjectures in A251416 are equivalent. At least to one side, our conjecture could be deduced from Klee's conjectures by a simple induction. - _Vladimir Shevelev_, Dec 10 2014
%F A251621 As a corollary, we have an explicit conjectural formula for prime(n), n>=8, essentially based on A098550: prime(n) = 19 + sum{i=9,...,n}a(i+4). - _Vladimir Shevelev_, Dec 11 2014
%e A251621 From _Vladimir Shevelev_, Dec 11 2014: (Start)
%e A251621 For formula for prime(n):
%e A251621 1) n=8, prime(8) = 19;
%e A251621 2) n=9, prime(9) = 19 + a(13) = 19 + 4 = 23;
%e A251621 3) n=10, prime(10)= 19 + a(13) + a(14) = 23 + 6 = 29, etc.
%e A251621 (End)
%t A251621 f[lst_] := Block[{k = 4}, While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 || MemberQ[lst, k], k++]; Append[lst, k]]; A098550 = Nest[f, {1, 2, 3}, 1000]; runningMax = Rest[FoldList[Max, -Infinity, #]]&; A249943 = runningMax[Take[Ordering[A098550], NestWhile[#+1&, 1, MemberQ[A098550, #] &] - 1]]; Length /@ Split[A249943] (* _Jean-François Alcover_, Sep 11 2017, using code from _Robert G. Wilson v_ *)
%o A251621 (Haskell)
%o A251621 import Data.List (group)
%o A251621 a251621 n = a251621_list !! (n-1)
%o A251621 a251621_list = map length $ group a249943_list
%Y A251621 Cf. A000040, A001223, A098550, A249943, A251416, A251620.
%K A251621 nonn
%O A251621 1,7
%A A251621 _Reinhard Zumkeller_, Dec 06 2014