A251554 a(1)=1, a(2)=2, a(3)=5; thereafter a(n) is the smallest number not occurring earlier having at least one common factor with a(n-2), but none with a(n-1).

1, 2, 5, 4, 15, 8, 3, 10, 9, 14, 27, 7, 6, 35, 12, 25, 16, 45, 22, 21, 11, 18, 55, 24, 65, 28, 13, 20, 39, 32, 33, 26, 51, 38, 17, 19, 34, 57, 40, 63, 44, 49, 30, 77, 36, 91, 46, 105, 23, 42, 115, 48, 85, 52, 75, 56, 69, 50, 81, 58, 93, 29, 31, 87, 62, 99, 64, 111

Offset: 1

Author

N. J. A. Sloane, Dec 21 2014

Keywords

• nonn

Comments

A variant of A098550. See that entry for much more information.

It seems likely that this sequence will never merge with A098550, but it would be nice to have a proof.

Links

• Chai Wah Wu, 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.

Crossrefs

Cf. A098550, A251555.

Programs

Haskell

import Data.List (delete)
a251554 n = a251554_list !! (n-1)
a251554_list = 1 : 2 : 5 : f 2 5 (3 : 4 : [6..]) 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 26 2014

Mathematica

a251554[lst_List] :=
Block[{k = 3},
  While[GCD[lst[[-2]], k] == 1 || GCD[lst[[-1]], k] > 1 ||
    MemberQ[lst, k], k++]; Append[lst, k]]; Nest[a251554, {1, 2,
5}, 120] (* Michael De Vlieger, Dec 23 2014, after Robert G. Wilson v at A098550 *)

Python

from math import gcd
A251554_list, l1, l2, s, b = [1,2,5], 5, 2, 3, {5}
for _ in range(10**4):
    i = s
    while True:
        if not i in b and gcd(i,l1) == 1 and gcd(i,l2) > 1:
            A251554_list.append(i)
            l2, l1 = l1, i
            b.add(i)
            while s in b:
                b.remove(s)
                s += 1
            break
        i += 1 # Chai Wah Wu, Dec 21 2014