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A349411
a(n) = prime j = A347113(i)-1 in order of appearance.
2
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
1,1
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, 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.
LINKS
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.
Eric Weisstein's World of Mathematics, Cunningham Chain.
EXAMPLE
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.
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, 239}]][[-1, -1]]; Select[m, PrimeQ]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Nov 16 2021
STATUS
approved