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

%I A349411 #14 Jun 20 2025 17:40:55
%S A349411 2,5,11,23,47,3,7,13,19,17,31,37,29,59,41,83,167,43,61,53,107,67,71,
%T A349411 73,79,89,179,359,719,1439,2879,97,101,103,109,113,227,131,263,127,
%U A349411 139,137,151,157,149,163,181,173,347,191,383,199,193,211,197,223,229,233
%N A349411 a(n) = prime j = A347113(i)-1 in order of appearance.
%C A349411 Let s = A347113, j = s(i)+1 and k = s(i+1). We recall the 3 constraints presented in A347113:
%C A349411 1. j = k is forbidden.
%C A349411 2. gcd(j,k) = 1 is forbidden.
%C A349411 3. All terms in s are distinct.
%C A349411 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.
%H A349411 Chris Caldwell's Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">Cunningham chains</a>.
%H A349411 Michael De Vlieger, <a href="/A349411/a349411.png">Log-log scatterplot of a(n)</a>, n=1..2^19, indicating prime j in red.
%H A349411 Michael De Vlieger, <a href="/A349411/a349411.txt">Extended table of n, a(n) for n=1..45450</a>
%H A349411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CunninghamChain.html">Cunningham Chain</a>.
%e A349411 s(1) = 1, thus j = s(1)+1 = 2, which is prime, therefore a(1) = 2.
%e A349411 s(2) = 4; j = 5, thus a(2) = 5, etc.
%t A349411 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]
%Y A349411 Cf. A347113, A348779, A349412.
%K A349411 nonn
%O A349411 1,1
%A A349411 _Michael De Vlieger_, Nov 16 2021