A239135 Numbers k such that (k-1)*k^2 + 1 and k^2 + (k-1) are both prime.

2, 3, 5, 6, 8, 13, 21, 24, 26, 28, 35, 45, 48, 50, 55, 76, 83, 89, 93, 96, 100, 101, 115, 120, 138, 140, 148, 149, 181, 191, 195, 203, 206, 209, 215, 230, 258, 259, 281, 285, 294, 301, 309, 323, 330, 349, 358, 373, 380, 386, 393, 395, 423, 428, 433, 474, 495

Offset: 1

Author

Ilya Lopatin following a suggestion from Juri-Stepan Gerasimov, Mar 15 2014

Keywords

• nonn

Comments

Numbers k such that (k^3 - k^2 + 1)*(k^2 + k - 1) is semiprime.

Intersection of A045546 and A111501.

Primes in this sequence: 2, 3, 5, 13, 83, 89, 101, 149, 181, 191, ...

Examples

			2 is in this sequence because (2-1)*2^2+1=5 and 2^2+(2-1)=5 are both prime.
		

Links

• Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A239115.

Programs

Magma

k := 1;
     for n in [1..10000] do
        if IsPrime(k*(n - 1)*n^2 + 1) and IsPrime(k*n^2 + n - 1) then
           n;
        end if;
     end for;

Mathematica

Select[Range[600],PrimeQ[#^2+#-1]&&PrimeQ[#^2(#-1)+1]&] (* Farideh Firoozbakht, Mar 17 2014 *)