A251620 -id:A251620 - cp's OEIS frontend

A249943 a(n) = smallest k such that the numbers 1..n appear among A098550(1), ..., A098550(k), or a(n) = 0 if there is no such k.

1, 2, 3, 4, 9, 10, 15, 15, 15, 16, 22, 22, 23, 23, 23, 23, 30, 31, 43, 43, 43, 43, 51, 51, 51, 51, 51, 51, 61, 61, 62, 62, 62, 62, 62, 62, 79, 79, 79, 79, 87, 87, 88, 88, 88, 88, 101

Offset: 1

Vladimir Shevelev, Dec 04 2014

nonn

The conjecture that all terms are positive is equivalent to the known conjecture that A098550 is a permutation of the positive integers.

Partial maxima of A098551: a(n) = max{a(n-1),A098551(n)} for n > 1. - Reinhard Zumkeller, Dec 06 2014

			Let n=6. Since A098550(9)=5 and A098550(10)=6, a(6)=10. - Corrected by _David Applegate_, Dec 08 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669, 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A098550, A098551.

Cf. A251620 (duplicates removed), A251621 (run lengths).

a249943 n = a249943_list !! (n-1)
a249943_list = scanl1 max $ map a098551 [1..]
-- Reinhard Zumkeller, Dec 06 2014

f[lst_List] := 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}, 100]; runningMax := Rest[FoldList[Max, -Infinity, #]]&; runningMax[Take[Ordering[A098550], NestWhile[#+1&, 1, MemberQ[A098550, #]&]-1]] (* Jean-François Alcover, Dec 05 2014, after Robert G. Wilson v and Peter J. C. Moses *)

The author conjectures that a(n)/n <= a(19)/19 = 43/19. Peter J. C. Moses verified that the strict inequality holds for 19 < n <= 1.1*10^5. - Vladimir Shevelev, Dec 06 2014

A251621 Run lengths in A249943.

Original entry on oeis.org

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, 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, 10, 14, 4, 2, 4, 14, 6, 10, 2, 4, 6, 8, 6, 6, 4, 6, 8, 4

Offset: 1

Reinhard Zumkeller, Dec 06 2014

nonn

			From _Vladimir Shevelev_, Dec 11 2014: (Start)
For formula for prime(n):
1) n=8, prime(8) = 19;
2) n=9, prime(9) = 19 + a(13) = 19 + 4 = 23;
3) n=10, prime(10)= 19 + a(13) + a(14) = 23 + 6 = 29, etc.
(End)

Bradley Klee, Posting to Sequence Fans Mailing List, Dec 07 2014
Vladimir Shevelev, Postings to Sequence Fans Mailing List, Dec 07, 10 and 11, 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
David L. Applegate, Hans Havermann, Bob Selcoe, Vladimir Shevelev, N. J. A. Sloane, and Reinhard Zumkeller, The Yellowstone Permutation, arXiv preprint arXiv:1501.01669 [math.NT], 2015 and J. Int. Seq. 18 (2015) 15.6.7.

Cf. A000040, A001223, A098550, A249943, A251416, A251620.

import Data.List (group)
a251621 n = a251621_list !! (n-1)
a251621_list = map length $ group a249943_list

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 *)

Connection with prime gaps: conjecturally, for n>=13, we have a(n) = A001223(n-5). - Vladimir Shevelev, Dec 07 2014
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
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

cp's OEIS Frontend

A249943 a(n) = smallest k such that the numbers 1..n appear among A098550(1), ..., A098550(k), or a(n) = 0 if there is no such k.

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