A309499 Primes p having a record value of least k such that 4*k^2*p^2 + 1 is prime.

2, 11, 17, 19, 283, 919, 1093, 1787, 9521, 181243, 257611, 274243, 857419, 1644871, 3111607, 6027277, 10452083, 14490703, 36102991, 47352131, 121431767, 171236887, 339934099, 584698243, 1177972427, 3008777311, 3091999399

Offset

1,1

Comments

The corresponding record values of k are 1, 3, 5, 15, 20, 22, 24, 45, 95, 104, 115, 116, 135, 143, 155, 165, 179, 186, 190, 245, 250, 260, 277, 284, 310, 313, 335, ...

Gagola calculated the values of k for all the primes below 5000, and noticed that the largest value of k was only 45.

Links

Table of n, a(n) for n=1..27.

Gloria Gagola, Progress on primes, News and Letters, Mathematics Magazine, Vol. 54, No. 1 (1981), p. 43.

Example

For the primes p = 2, 3, 5, and 7, 4*p^2 + 1 = 17, 37, 101, and 197 are all primes with k = 1. 11 is the first prime with a value of k = 3, since 4*1*11^2 + 1 = 45 and 4*2^2*11^2 + 1 = 1937 are both composites, and 4*3^2*11^2 + 1 = 4357 is prime.

Mathematica

a[p_] := Module[{k = 1}, While[!PrimeQ[4 * k^2 * p^2 + 1], k++]; k]; s={}; am = 0; p = 1; Do[p = NextPrime[p]; a1 = a[p]; If[a1 > am, am=a1; AppendTo[s, p]], {n, 1, 20000}]; s

Crossrefs

Cf. A309498.

Sequence in context: A019402 A038927 A105877 * A173638 A018420 A253474

Adjacent sequences: A309496 A309497 A309498 * A309500 A309501 A309502

Keyword

nonn,more

Author

Amiram Eldar, Aug 05 2019

Status

approved