A357595 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1; j = n + a(n).

1, 4, 2, 10, 6, 22, 7, 8, 12, 3, 26, 74, 14, 9, 46, 122, 15, 16, 17, 18, 19, 5, 21, 11, 20, 24, 25, 13, 82, 27, 30, 183, 35, 28, 31, 32, 34, 142, 33, 36, 38, 158, 40, 166, 39, 42, 44, 49, 194, 45, 50, 202, 48, 303, 51, 52, 54, 37, 55, 56, 29, 57, 63, 58, 60, 65

Offset: 1

Author

David James Sycamore, Oct 05 2022

Keywords

• nonn

Comments

If n + a(n) = prime p, a(n+1) is the smallest multiple (>1) of p, which has not occurred earlier. Conjectured to be a permutation of the positive integers.

Examples

			a(1)=1, then 1+a(1)=2 so a(2) must be 4, the least k != 2 which shares a divisor with 2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16384
• Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^14, labeling records in red and local minima in blue, highlighting primes in green and (composite) prime powers in gold.

Crossrefs

Cf. A347113, A349472.

Programs

Mathematica

nn = 66; c[] = False; u = 2; a[1] = j = 1; c[1] = True; Do[Set[{k, m}, {u, n + j - 1}]; While[Or[c[k], k == m, CoprimeQ[k, m]], k++]; Set[{a[n], c[k], j}, {k, True, k}]; If[k == u, While[c[u], u++]], {n, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, Oct 05 2022 *)