%I #21 Jan 24 2021 18:32:11

%S 11,29,438926021,1210400879,7446335849,31757068151,33090566651,

%T 33164857769,40137398219,45133754591,46642404071,100444384301,

%U 114546675671,144553207691,159587584529,161557039991,166054101539,210447830009,227625400031,236241327599,254850262949,272259344081

%N Primes p such that p + 2*k^2 is prime for at least 10 consecutive values of k starting from k=1.

%C Number 10 was chosen as a threshold as the smallest two digit number. You can choose other numbers and if they are less than 12, the first terms of sequences analogous to this one will be those in A165234.

%C There are 20 primes like that among the first 10^10 of them. The second term, 29, generates 28 primes (A007641). Sixteen others, including 11 (A050265), generate only 10 primes, while three produce 11 primes. These three are: 33164857769 (see also A165234), 159587584529, and 236241327599. The first term among the second 10^10 of primes is 254850262949. Then there is 272259344081 (mentioned in A165234) that generates 13 primes.

%C All these primes end with 1 or 9 and are congruent to 5 mod 6.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html">Prime-generating Polynomial</a>

%e 11 is a term because 11+2*k^2 gives rise to 10 primes for 10 consecutive values of k starting from 1 (see A050265).

%t lst={}; Do[k=1; While[PrimeQ[Prime[n]+2*k^2], k++]; If[k>10, AppendTo[lst, Prime[n]]], {n, 2, 11*10^9}]; lst

%t Select[Prime[Range[107669*10^5]],AllTrue[#+{2,8,18,32,50,72,98,128,162,200},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* The program will take a long time to run *) (* _Harvey P. Dale_, Jan 24 2021 *)

%o (PARI) forprime(n=2, 276241327599, k=1; while(isprime(n+2*k^2), k++); (k>10)&&print1(n, ", "))

%Y Cf. A000040 (primes), A050265, A007641, A271366, A271818, A271819, A271820 (examples of sequences of primes generated by terms of this sequence), A165234.

%K nonn

%O 1,1

%A _Waldemar Puszkarz_, Apr 04 2016