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Positions of prime j = A347113(i)+1.
2

%I #9 Apr 03 2023 10:36:13

%S 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 88,89,90,91,92,93,102,105,108,113,121,122,126,127,133,140,143,149,

%U 156,160,167,180,185,186,191,192,199,203,208,211,231,235,240,241,245

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

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

%C 1. j = k is forbidden.

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

%C 3. All terms in s are distinct.

%C 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 Chris Caldwell's Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=CunninghamChain">Cunningham chains</a>.

%H Michael De Vlieger, <a href="/A349412/a349412.txt">Extended table of n, a(n) for n = 1..45450</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CunninghamChain.html">Cunningham Chain</a>.

%e s(1) = 1, thus j = s(1)+1 = 2, which is prime, therefore a(1) = 1.

%e s(2) = 4; j = 5, thus a(2) = 2, etc.

%t 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 Cf. A347113, A347313, A349411.

%K nonn

%O 1,2

%A _Michael De Vlieger_, Nov 16 2021