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 A309499 Primes p having a record value of least k such that 4*k^2*p^2 + 1 is prime. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified April 9 10:52 EDT 2020. Contains 333348 sequences. (Running on oeis4.)