A349412 -id:A349412 - cp's OEIS frontend

A349411 a(n) = prime j = A347113(i)-1 in order of appearance.

2, 5, 11, 23, 47, 3, 7, 13, 19, 17, 31, 37, 29, 59, 41, 83, 167, 43, 61, 53, 107, 67, 71, 73, 79, 89, 179, 359, 719, 1439, 2879, 97, 101, 103, 109, 113, 227, 131, 263, 127, 139, 137, 151, 157, 149, 163, 181, 173, 347, 191, 383, 199, 193, 211, 197, 223, 229, 233

Offset: 1

Michael De Vlieger, Nov 16 2021

nonn

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, since gcd(j,k)=1 and j=k is forbidden. In the context of s, j | k implies j < k and sequence increase. The least k > j such that j | k is 2j, giving rise to Cunningham chains of the first kind.

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

Chris Caldwell's Prime Glossary, Cunningham chains.
Michael De Vlieger, Log-log scatterplot of a(n), n=1..2^19, indicating prime j in red.
Michael De Vlieger, Extended table of n, a(n) for n=1..45450
Eric Weisstein's World of Mathematics, Cunningham Chain.

Cf. A347113, A348779, A349412.

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, 239}]][[-1, -1]]; Select[m, PrimeQ]

cp's OEIS Frontend

A349411 a(n) = prime j = A347113(i)-1 in order of appearance.

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