A103683 a(1)=1, a(2)=2, a(3)=3, a(n) = smallest positive integer not occurring earlier in sequence and coprime to a(n-1), a(n-2) and a(n-3).

1, 2, 3, 5, 7, 4, 9, 11, 13, 8, 15, 17, 19, 14, 23, 25, 27, 16, 29, 31, 21, 10, 37, 41, 33, 20, 43, 47, 39, 22, 35, 53, 51, 26, 49, 55, 57, 32, 59, 61, 45, 28, 67, 71, 65, 6, 73, 77, 79, 12, 83, 85, 89, 18, 91, 95, 97, 24, 101, 103, 107, 30, 109, 113, 119, 36, 115, 121, 127, 34

Offset: 1

Author

Leroy Quet, Mar 26 2005

Keywords

• nonn

Comments

Conjectured to be not a permutation of the natural numbers.

Charles R Greathouse IV extended this, and confirms that primes occur in natural order. - Jonathan Vos Post and M. F. Hasler, Jan 18 2011

Conjecture: for n >= 67, a(n) is even if and only if n == 2 (mod 4) and divisible by 3 if and only if n == 3 (mod 4). In particular, this implies the last value divisible by 6 is a(66) = 36. - Robert Israel, May 12 2015

a(102982) = 42, see A105214. Conjecture above is false. - Sergio Pimentel, Apr 18 2022

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A084937, A105214, A352950.

Programs

Maple

ina:= proc(n) false end:
a:= proc(n) option remember; local k;
      if n<4 then k:= n
    else for k from 4 while ina(k) or igcd(k, a(n-1))<>1 or
                igcd(k, a(n-2))<>1 or igcd(k, a(n-3))<>1
         do od
      fi; ina(k):= true; k
    end:
seq(a(n), n=1..120);  # Alois P. Heinz, Jan 19 2011

Mathematica

f[s_] := Block[{k = 1, l = Take[s, -3]}, While[ Union[ GCD[k, l]] != {1} || MemberQ[s, k], k++]; Append[s, k]]; Nest[f, {1, 2, 3}, 70] (* Robert G. Wilson v, Jun 26 2011 *)

Python

from math import gcd
from itertools import islice
def agen(): # generator of terms
    aset, b, c, d = {1, 2, 3, 5}, 2, 3, 5
    yield from [1, b, c, d]
    while True:
        k = 1
        while k in aset or any(gcd(t, k) != 1 for t in [b, c, d]): k+= 1
        b, c, d = c, d, k
        aset.add(k)
        yield k
print(list(islice(agen(), 70))) # Michael S. Branicky, Apr 18 2022

Extensions

More terms from Robert G. Wilson v, Mar 30 2005