A321869 Numbers k such that m = 3k^2 + 2k + 10 and 3m - 2 are both primes.

3, 9, 17, 57, 69, 177, 293, 303, 317, 339, 377, 407, 429, 437, 443, 467, 503, 573, 597, 759, 783, 797, 849, 897, 1329, 1343, 1409, 1899, 1923, 2267, 2357, 2427, 2579, 2679, 2739, 2843, 2967, 3089, 3263, 3279, 3303, 3323, 3419, 3459, 3509, 3933, 3999, 4293

Offset: 1

Author

Amiram Eldar, Nov 20 2018

Keywords

• nonn

Comments

Rotkiewicz proved that if k is in this sequence, and m = 3k^2 + 2k + 10, then m*(3m - 2) is an octagonal Fermat pseudoprime to base 2 (A321868), and thus under Schinzel's Hypothesis H there are infinitely many octagonal Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 467285, 1532245, 20134661, 26190165, 52685061, 95519061, ...

Examples

			3 is in the sequence since 3*3^2 + 2*3 + 10 = 43 and 3*43 - 2 = 127 are both primes.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
• Wikipedia, Schinzel's Hypothesis H.

Crossrefs

Cf. A000567, A001567, A321868.

Programs

Mathematica

Select[Range[1000], PrimeQ[3#^2 + 2# + 10] && PrimeQ[9#^2 + 6# + 28]  &]
Select[Range[4300],With[{m=3#^2+2#+10},AllTrue[{m,3m-2},PrimeQ]&]] (* Harvey P. Dale, Dec 13 2024 *)

PARI

isok(n) = isprime(m=3*n^2 + 2*n + 10) && isprime(3*m-2); \\ Michel Marcus, Nov 20 2018