A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 73, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 89, 91, 97, 100

Offset

1,2

Comments

The quadratic polynomial p(k) = k^2 + 21*k + 1 is not a prime-generating polynomial in the sense of Eric Weisstein's World of Mathematics (see link) because p(0) is not prime.

However p(k) is prime for the first 17 positive integral values of k and among polynomials of the form k^2 + j*k + 1, the present polynomial (j = 21) generates more primes than any polynomial of that form for any positive integral j < 231, at least for positive integers, k, in the range 0 < k < 10^6.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Example

17 is in the sequence because 17^2 + 21 * 17 + 1 = 647 is prime.
18 is not in the sequence because 18^2 + 21 * 18 + 1 = 703 = 19 * 37.

Maple

select(k-> isprime(k^2+21*k+1), [$1..100])

Mathematica

Select[Range[100], PrimeQ[#^2 + 21# + 1] &] (* Alonso del Arte, Mar 06 2018 *)

Prog

(PARI) isok(k) = isprime(k^2+21*k+1); \\ Altug Alkan, Mar 07 2018

Crossrefs

Cf. A292509, A050267, A050268, A005846, A007635, A007641, A048988.

Sequence in context: A342523 A367615 A249830 * A271980 A353886 A191882

Adjacent sequences: A300470 A300471 A300472 * A300474 A300475 A300476

Keyword

nonn

Author

James R. Buddenhagen, Mar 06 2018

Status

approved