A357614 Lexicographically earliest infinite sequence of distinct positive integers such that a(n+1) is the least k != j, for which gcd(k, j) > 1, where j = a(n) + prime(n).

1, 6, 3, 2, 12, 46, 118, 5, 4, 9, 8, 13, 10, 15, 14, 122, 7, 11, 16, 166, 18, 21, 20, 206, 25, 22, 24, 254, 19, 26, 278, 27, 28, 30, 39, 32, 33, 34, 394, 17, 35, 36, 31, 37, 23, 38, 42, 44, 45, 40, 538, 48, 41, 47, 50, 614, 1754, 49, 52, 56, 674, 29, 54, 57, 58

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..65.

Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, labeling maxima in red and local minima in blue, highlighting primes in green and other prime powers in gold.

Example

a(1)=1; 1 + prime(1)=3, so a(2) = 6, the smallest unused number sharing a divisor with 3.

Mathematica

nn = 65; c[_] = False; u = 2; a[1] = j = 1; c[1] = True; Do[m = j + Prime[n - 1]; If[PrimeQ[m], k = 2; While[c[k m], k++]; k *= m, k = u; 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 06 2022 *)

Crossrefs

Cf. A347113, A349472, A357595.

Sequence in context: A114348 A280680 A272083 * A096840 A096685 A307150

Adjacent sequences: A357611 A357612 A357613 * A357615 A357616 A357617

Keyword

nonn

Author

David James Sycamore, Oct 06 2022

Status

approved