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A306400
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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.
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2
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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
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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