A349412 Positions of prime j = A347113(i)+1.

1, 2, 3, 4, 5, 8, 9, 13, 17, 21, 25, 31, 34, 35, 39, 40, 41, 45, 56, 60, 61, 66, 70, 75, 79, 88, 89, 90, 91, 92, 93, 102, 105, 108, 113, 121, 122, 126, 127, 133, 140, 143, 149, 156, 160, 167, 180, 185, 186, 191, 192, 199, 203, 208, 211, 231, 235, 240, 241, 245

Offset: 1

Author

Michael De Vlieger, Nov 16 2021

Keywords

• nonn

Comments

Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:

1. j = k is forbidden.

2. gcd(j,k) = 1 is forbidden.

3. All terms in s are distinct.

These constraints confine prime j to the relationship j | k, which in the context of s, implies j < k and increase. The least k > j such that j | k is 2j, giving rise to Cunningham chains of the first kind. The chains are evident in this sequence as series of consecutive positions in s.

Examples

			s(1) = 1, thus j = s(1)+1 = 2, which is prime, therefore a(1) = 1.
s(2) = 4; j = 5, thus a(2) = 2, etc.
		

Links

• Chris Caldwell's Prime Glossary, Cunningham chains.
• Michael De Vlieger, Extended table of n, a(n) for n = 1..45450
• Eric Weisstein's World of Mathematics, Cunningham Chain.

Crossrefs

Cf. A347113, A347313, A349411.

Programs

Mathematica

c[] = 0; j = m = 2; m = 1 + {1}~Join~Reap[Do[If[IntegerQ@ Log2[i], While[c[m] > 0, m++]]; Set[k, m]; While[Or[c[k] > 0, k == j, GCD[j, k] == 1], k++]; Sow[k]; Set[c[k], i]; j = k + 1, {i, 245}]][[-1, -1]]; Position[m, ?PrimeQ][[All, 1]]