This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A349412 #13 Jun 20 2025 17:40:27
%S A349412 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,
%T A349412 88,89,90,91,92,93,102,105,108,113,121,122,126,127,133,140,143,149,
%U A349412 156,160,167,180,185,186,191,192,199,203,208,211,231,235,240,241,245
%N A349412 Positions of prime j = A347113(i)+1.
%C A349412 Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:
%C A349412 1. j = k is forbidden.
%C A349412 2. gcd(j,k) = 1 is forbidden.
%C A349412 3. All terms in s are distinct.
%C A349412 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.
%H A349412 Chris Caldwell's Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">Cunningham chains</a>.
%H A349412 Michael De Vlieger, <a href="/A349412/a349412.txt">Extended table of n, a(n) for n = 1..45450</a>
%H A349412 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CunninghamChain.html">Cunningham Chain</a>.
%e A349412 s(1) = 1, thus j = s(1)+1 = 2, which is prime, therefore a(1) = 1.
%e A349412 s(2) = 4; j = 5, thus a(2) = 2, etc.
%t 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, 245}]][[-1, -1]]; Position[m, _?PrimeQ][[All, 1]]
%Y A349412 Cf. A347113, A347313, A349411.
%K A349412 nonn
%O A349412 1,2
%A A349412 _Michael De Vlieger_, Nov 16 2021