A361929 a(1) = 2; for n > 1, a(n) is the smallest positive integer > 1 not to share a factor with terms a(n-c .. n-1) where c = gcd(n-1,a(n-1)).

2, 3, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 11, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 11, 2, 3, 2, 3, 5, 2, 3, 7, 2, 3, 5, 2, 3, 7, 11, 2, 3, 2, 5, 7, 2, 3, 5, 2, 3, 7, 2, 3, 5, 11

Offset: 1

Author

Samuel Harkness, Mar 30 2023

Keywords

• nonn

Comments

Conjecture: All primes will occur. See A361931 for the first occurrences of primes.

a(p+1) = 2 or 3 when p is prime.

Examples

			For a(26), we see a(25) = 5. Then gcd(25,5) = 5, so a(26) must not share a factor with any of the previous 5 terms. The previous 5 terms {a(21), a(22), a(23), a(24), a(25)} = {3, 7, 2, 3, 5}, and the least positive number not to share a factor with {3, 7, 2, 3, 5} is 11, so a(26) = 11.
The first terms, alongside gcd(n,a(n)):
  n  a(n)  gcd(n,a(n))
  -  ----  ----
  1     2     1
  2     3     1
  3     2     1
  4     3     1
  5     2     1
  6     3     3
  7     5     1
  8     2     2
  9     3     3
  10    7     1
		

Links

• Samuel Harkness, Table of n, a(n) for n = 1..13812

Crossrefs

Cf. A358921, A361931 (indices of first occurrences).

Programs

Mathematica

K = {2}; While[Length@K < 86, p = 2; While[MemberQ[K[[Length@K - GCD[Length@K, Last@K] + 1 ;; Length@K]], p], p = NextPrime[p]]; AppendTo[K, p]]; Print[K]

PARI

isok(w, k) = for (i=1, #w, if (gcd(k, w[i]) > 1 , return(0));); 1;
lista(nn) = my(va = vector(nn)); va[1] = 2; for (n=2, nn, my(k=2, ok = 0, w = vector(gcd(n-1, va[n-1]), i, va[n-i])); while (!ok, ok = isok(w, k); if (!ok, k++);); va[n] = k;); va; \\ Michel Marcus, Mar 31 2023