

A306400


For the nth prime p of the form 6k1, a(n) is the first prime q for which (p+q^21, p+q^2+1) is a twin prime pair.


0



5, 7, 5, 7, 11, 23, 5, 7, 7, 11, 5, 7, 7, 11, 5, 7, 701, 7, 5, 5, 7, 7, 41, 11, 7, 19, 13, 5, 7, 31, 17, 13, 11, 31, 41, 13, 31, 7, 29, 11, 37, 13, 53, 11, 19, 19, 11, 13, 23, 37, 7, 41, 23, 7, 29, 5, 71, 5, 13, 29, 13, 13, 59, 97, 11, 37, 17, 7, 7, 5, 7, 157
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OFFSET

1,1


COMMENTS

Actually, there is no need to test for q=2 and q=3, as 6k1+4+1 = (6k+2, 6k+4), both terms not prime; and 6k1+9+1 = (6k+7,6k+9), with 6k+9 not prime.
The sequence could be extended to nonprime numbers p=6k1 and/or nonprime numbers q=6t+1. However, it could not be extended to p=6k+1 (prime or not), because for q=6t+1, p+q^2 = 6k+1+36t^2+12t+1 ≡ 2 (mod 6); hence p+q^2+1 == 3 (mod 6) is never a prime number.
Proving that this sequence is infinite would prove the twin prime conjecture (that there are infinitely many twin primes), as the twin prime pair associated with the prime p is greater than p and there are infinitely many prime numbers.
This sequence refers to the first q for which p+q^2+1 is a twin prime pair. However, analysis (by computer program) suggests that for each prime p there are infinitely many primes q for which p+q^2+1 is a twin prime pair. Proving this statement, even for a single prime p, would prove the twin prime conjecture.


LINKS

Table of n, a(n) for n=1..72.


EXAMPLE

Example: for n=5, a(5) = 11, because the 5th prime of the form 6k1 is 29, and 29+11^2+1 = (149,151) is a twin prime pair, while 29+2^2+1, 29+3^2+1, 29+5^2+1, 29+7^2+1 are not twin prime pairs.


MATHEMATICA

Table[Block[{q = 2}, While[! AllTrue[p + q^2 + {1, 1}, PrimeQ], q = NextPrime@ q]; q], {p, Select[Range[5, 825, 6], PrimeQ]}] (* Michael De Vlieger, Mar 31 2019 *)


PROG

(PARI) lista(nn) = {forprime(p=2, nn, if (((p+1) % 6) == 0, my(q=5); while (!(isprime(p+q^21) && isprime(p+q^2+1)), q = nextprime(q+1)); print1(q, ", "); ); ); } \\ Michel Marcus, Mar 26 2019


CROSSREFS

Cf. A007528, A001359 (lesser of twin primes).
Sequence in context: A215732 A010718 A247872 * A090987 A278813 A217167
Adjacent sequences: A306397 A306398 A306399 * A306401 A306402 A306403


KEYWORD

nonn,easy


AUTHOR

Paolo Iachia, Mar 25 2019


STATUS

approved



