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 A251413 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). 3
 1, 3, 5, 9, 25, 21, 55, 7, 11, 35, 33, 49, 15, 77, 27, 91, 45, 13, 51, 65, 17, 39, 85, 57, 115, 19, 23, 95, 69, 125, 63, 145, 81, 29, 75, 203, 93, 119, 31, 105, 341, 87, 121, 111, 143, 37, 99, 185, 117, 155, 123, 175, 41, 133 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture 1: Sequence is a permutation of the odd numbers. Conjecture 2: The odd primes occur in the sequence in their natural order. Comments from N. J. A. Sloane, Dec 13 2014: (Start) The following properties are known (the proofs are analogous to the proofs for the corresponding facts about A098550). 1. The sequence is infinite. 2. At least one-third of the terms are composite. 3. For any odd prime p, there is a term that is divisible by p. 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. (End) REFERENCES L. Edson Jeffery, Posting to Sequence Fans Mailing List, Dec 01 2014 LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..11945 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. MAPLE N:= 10^3: # to get a(1) to a(n) where a(n+1) is the first term > N B:= Vector(N, datatype=integer[4]): for n from 1 to 3 do A[n]:= 2*n-1: od: for n from 4 do for k from 4 to N do if B[k] = 0 and igcd(2*k-1, A[n-1]) = 1 and igcd(2*k-1, A[n-2]) > 1 then A[n]:= 2*k-1; B[k]:= 1; break fi od: if 2*k-1 > N then break fi od: seq(A[i], i=1..n-1); # Based on Robert Israel's program for A098550 MATHEMATICA 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 *) PROG (Python) from fractions import gcd A251413_list, l1, l2, s, b = [1, 3, 5], 5, 3, 7, {} for _ in range(1, 10**4): ....i = s ....while True: ........if not i in b and gcd(i, l1) == 1 and gcd(i, l2) > 1: ............A251413_list.append(i) ............l2, l1, b[i] = l1, i, True ............while s in b: ................b.pop(s) ................s += 2 ............break ........i += 2 # Chai Wah Wu, Dec 07 2014 (Haskell) import Data.List (delete) a251413 n = a251413_list !! (n-1) a251413_list = 1 : 3 : 5 : f 3 5 [7, 9 ..] where f u v ws = g ws where g (x:xs) = if gcd x u > 1 && gcd x v == 1 then x : f v x (delete x ws) else g xs -- Reinhard Zumkeller, Dec 25 2014 CROSSREFS Cf. A098550, A251414. Sequence in context: A262483 A083366 A006722 * A039774 A114001 A306838 Adjacent sequences: A251410 A251411 A251412 * A251414 A251415 A251416 KEYWORD nonn AUTHOR N. J. A. Sloane, Dec 02 2014 STATUS approved

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Last modified January 29 01:41 EST 2023. Contains 359905 sequences. (Running on oeis4.)