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A306400 For the n-th prime p of the form 6k-1, a(n) is the first prime q for which (p+q^2-1, p+q^2+1) is a twin prime pair. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Actually, there is no need to test for q=2 and q=3, as 6k-1+4+-1 = (6k+2, 6k+4), both terms not prime; and 6k-1+9+-1 = (6k+7,6k+9), with 6k+9 not prime.

The sequence could be extended to nonprime numbers p=6k-1 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

Robert Israel, Table of n, a(n) for n = 1..10000

EXAMPLE

Example: for n=5, a(5) = 11, because the 5th prime of the form 6k-1 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.

MAPLE

g:= proc(p) local q;

  q:= 3:

  do

    q:= nextprime(q);

    if isprime(p+q^2-1) and isprime(p+q^2+1) then return q fi;

  od

end proc:

map(g, select(isprime, [seq(i, i=5..1000, 6)])); # Robert Israel, Nov 23 2020

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^2-1) && 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

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Last modified December 5 23:38 EST 2021. Contains 349558 sequences. (Running on oeis4.)