A292509 Primes of the form k^2 + 23*k + 23.

23, 47, 73, 101, 131, 163, 197, 233, 271, 311, 353, 397, 443, 491, 541, 593, 647, 761, 821, 883, 947, 1013, 1151, 1223, 1297, 1373, 1451, 1531, 1613, 1697, 1783, 1871, 2053, 2243, 2341, 2441, 2543, 2647, 2753, 2861, 2971, 3083, 3313, 3673, 3797, 3923, 4051, 4447

Offset: 1

Author

Waldemar Puszkarz, Sep 17 2017

Keywords

• nonn

Comments

The first 17 primes correspond to n from 0 to 16, which makes n^2 + 23n + 23 a prime-generating polynomial (see the link). This is a monic polynomial of the form n^2 + pn + p, where p is prime. Among the first 10^8 primes, only two more besides 23 give rise to prime-generating polynomials of this form. They are 8693 and 50983511 and they generate only 11 primes for n = 0 to 10.

Examples

			For n = 1, we have 1^2 + 23 * 1 + 23 = 47, which is prime, so 47 is in the sequence.
For n = 2, we have 2^2 + 23 * 2 + 23 = 4 + 46 + 23 = 73, which is prime, so 73 is in the sequence.
Contrast to n = 17, which gives us 17^2 + 23 * 17 + 23 = 289 + 391 + 23 = 703 = 19 * 37, so 703 is not in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Crossrefs

Cf. A005846, A300473.

Programs

Magma

[a: n in [0..100] | IsPrime(a) where a is n^2+23*n+23 ]; // Vincenzo Librandi, Sep 23 2017

Maple

select(isprime, [seq(x^2+23*x+23, x=0..1000)]); # Robert Israel, Sep 18 2017

Mathematica

Select[Range[0, 100]//#^2 + 23# + 23 &, PrimeQ]

PARI

for(n=0, 100, isprime(n^2+23*n+23)&&print1(n^2+23*n+23 ","))