A271348 Primes p such that p + 2*k^2 is prime for at least 10 consecutive values of k starting from k=1.

11, 29, 438926021, 1210400879, 7446335849, 31757068151, 33090566651, 33164857769, 40137398219, 45133754591, 46642404071, 100444384301, 114546675671, 144553207691, 159587584529, 161557039991, 166054101539, 210447830009, 227625400031, 236241327599, 254850262949, 272259344081

Offset: 1

Author

Waldemar Puszkarz, Apr 04 2016

Keywords

• nonn

Comments

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.

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.

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

Examples

			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).
		

Links

• Eric Weisstein's World of Mathematics, Prime-generating Polynomial

Crossrefs

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

Programs

Mathematica

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
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 *)

PARI

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