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

1,1

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, ...

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000567, A001567, A321868.

Sequence in context: A018307 A108050 A343781 * A009211 A105538 A374692

Adjacent sequences: A321866 A321867 A321868 * A321870 A321871 A321872

Keyword

nonn

Author

Amiram Eldar, Nov 20 2018

Status

approved