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A334287 Smallest full reptend prime p such that there is a gap of exactly 2n between p and the next full reptend prime, or 0 if no such prime exists.

%I A334287 #15 Feb 16 2025 08:34:00
%S A334287 17,19,23,491,7,47,419,577,29,0,1789,233,461,433,193,509,823,61,1979,
%T A334287 1327,659,269,11503,1381,887,14251,3167,8297,3469,0,7247,15073,2473,
%U A334287 743,19309,4349,21503,12823,14939,3863,5419,6389,24137,27211,10343,13577,18979
%N A334287 Smallest full reptend prime p such that there is a gap of exactly 2n between p and the next full reptend prime, or 0 if no such prime exists.
%C A334287 Gaps of length congruent to 20 mod 40 do not exist. All full reptend primes are either 7, 11, 17, 19, 21, 23, 29, or 33 mod 40, and no difference of 20 exists between any of these numbers.
%H A334287 Martin Raab, <a href="/A334287/b334287.txt">Table of n, a(n) for n = 1..583</a>
%H A334287 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FullReptendPrime.html">Full Reptend Prime</a>
%e A334287 a(9) = 29 because there is a gap of 2*9 = 18 between 29 and the next full reptend prime 47.
%e A334287 a(10) = 0 because no gap of 2*10 = 20 exists between full reptend primes.
%o A334287 (PARI) is(p) = Mod(10, p)^(p\2)==-1 && znorder(Mod(10, p))+1==p;
%o A334287 isok(p, n) = {if (! is(p), return (0)); if (isprime(p+n) && is(p+n), forprime(q=p+1, p+n-1, if (is(q), return (0));); return (1););}
%o A334287 a(n) = {n *= 2; if ((n % 40) == 20, return (0)); my (p = 2); while (! isok(p, n), p = nextprime(p+1)); p;} \\ _Michel Marcus_, Apr 22 2020
%Y A334287 Cf. A001913.
%K A334287 nonn
%O A334287 1,1
%A A334287 _Martin Raab_, Apr 21 2020