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%I #38 May 15 2018 16:41:52
%S 1,3,5,9,25,21,55,7,11,35,33,49,15,77,27,91,45,13,51,65,17,39,85,57,
%T 115,19,23,95,69,125,63,145,81,29,75,203,93,119,31,105,341,87,121,111,
%U 143,37,99,185,117,155,123,175,41,133
%N a(n) = 2n-1 if n <= 3, otherwise the smallest odd number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1).
%C Conjecture 1: Sequence is a permutation of the odd numbers.
%C Conjecture 2: The odd primes occur in the sequence in their natural order.
%C Comments from _N. J. A. Sloane_, Dec 13 2014: (Start)
%C The following properties are known (the proofs are analogous to the proofs for the corresponding facts about A098550).
%C 1. The sequence is infinite.
%C 2. At least one-third of the terms are composite.
%C 3. For any odd prime p, there is a term that is divisible by p.
%C 4. Let a(n_p) be the first term that is divisible by p. Then a(n_p) = q*p where q is an odd prime less than p. If p < r are primes then n_p < n_r.
%C (End)
%D L. Edson Jeffery, Posting to Sequence Fans Mailing List, Dec 01 2014
%H N. J. A. Sloane, <a href="/A251413/b251413.txt">Table of n, a(n) for n = 1..11945</a>
%H 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, 2015 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Sloane/sloane9.html">J. Int. Seq. 18 (2015) 15.6.7</a>.
%p N:= 10^3: # to get a(1) to a(n) where a(n+1) is the first term > N
%p B:= Vector(N, datatype=integer[4]):
%p for n from 1 to 3 do A[n]:= 2*n-1: od:
%p for n from 4 do
%p for k from 4 to N do
%p if B[k] = 0 and igcd(2*k-1, A[n-1]) = 1 and igcd(2*k-1, A[n-2]) > 1 then
%p A[n]:= 2*k-1;
%p B[k]:= 1;
%p break
%p fi
%p od:
%p if 2*k-1 > N then break fi
%p od:
%p seq(A[i], i=1..n-1); # Based on _Robert Israel_'s program for A098550
%t max = 54; f = True; a = {1, 3, 5}; NN = Range[4, 1000]; s = 2*NN - 1; While[TrueQ[f], For[k = 1, k <= Length[s], k++, If[Length[a] < max, If[GCD[a[[-1]], s[[k]]] == 1 && GCD[a[[-2]], s[[k]]] > 1, a = Append[a, s[[k]]]; s = Delete[s, k]; k = 0; Break], f = False]]]; a (* _L. Edson Jeffery_, Dec 02 2014 *)
%o (Python)
%o from fractions import gcd
%o A251413_list, l1, l2, s, b = [1,3,5], 5, 3, 7, {}
%o for _ in range(1,10**4):
%o ....i = s
%o ....while True:
%o ........if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
%o ............A251413_list.append(i)
%o ............l2, l1, b[i] = l1, i, True
%o ............while s in b:
%o ................b.pop(s)
%o ................s += 2
%o ............break
%o ........i += 2 # _Chai Wah Wu_, Dec 07 2014
%o (Haskell)
%o import Data.List (delete)
%o a251413 n = a251413_list !! (n-1)
%o a251413_list = 1 : 3 : 5 : f 3 5 [7, 9 ..] where
%o f u v ws = g ws where
%o g (x:xs) = if gcd x u > 1 && gcd x v == 1
%o then x : f v x (delete x ws) else g xs
%o -- _Reinhard Zumkeller_, Dec 25 2014
%Y Cf. A098550, A251414.
%K nonn
%O 1,2
%A _N. J. A. Sloane_, Dec 02 2014
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