A165234 Least prime p such that 2x^2 + p produces primes for x=0..n-1 and composite for x=n.

2, 17, 3, 1481, 5, 149, 569, 2081, 2339, 5939831, 11, 33164857769, 3217755097229, 272259344081, 17762917045631

Offset: 1

Author

T. D. Noe, Sep 09 2009

Keywords

• hard
• nonn
• more

Comments

Other known values: a(14)=272259344081 and a(29)=29. There are no other terms less than 10^12. The primes p = 3, 5, 11, and 29 produce p consecutive distinct primes because the imaginary quadratic field Q(sqrt(-2p)) has class number 2. Assuming the prime k-tuples conjecture, this sequence is defined for n>0.

References

• Paulo Ribenboim, My Numbers, My Friends, Springer,2000, pp. 349-350.

Links

• R. A. Mollin, Prime-producing quadratics, Amer. Math. Monthly 104 (1997), 529-544.
• Eric W. Weisstein, Prime-Generating Polynomial

Crossrefs

Cf. A007641, A050265, A161008, A164926.

Programs

Mathematica

PrimeRun[p_Integer] := Module[{k=0}, While[PrimeQ[2k^2+p], k++ ]; k]; nn=9; t=Table[0,{nn}]; cnt=0; p=1; While[cnt
				

PARI

isok(p, n) = for (k=0, n-1, if(!isprime(p + 2*k^2), return(0))); return(!isprime(p + 2*n^2));
a(n) = forprime(p=2, oo, if(isok(p, n), return(p))); \\ Daniel Suteu, Dec 22 2024

Perl

use ntheory qw(:all); sub a { my $n = $[0]; my $lo = 2; my $hi = 2*$lo; while (1) { my @terms = grep { !is_prime($ + 2*$n*$n) } sieve_prime_cluster($lo, $hi, map { 2*$*$ } 1 .. $n-1); return $terms[0] if @terms; $lo = $hi+1; $hi = 2*$lo; } }; $| = 1; for my $n (1..100) { print a($n), ", " } # Daniel Suteu, Dec 22 2024

Extensions

a(13) and a(15) from Daniel Suteu, Dec 22 2024