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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.

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 A171879

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 April 23 18:45 EDT 2017. Contains 285329 sequences.