A321869 - OEIS

%I #15 Jul 17 2021 06:55:15

%S 3,9,17,57,69,177,293,303,317,339,377,407,429,437,443,467,503,573,597,

%T 759,783,797,849,897,1329,1343,1409,1899,1923,2267,2357,2427,2579,

%U 2679,2739,2843,2967,3089,3263,3279,3303,3323,3419,3459,3509,3933,3999,4293

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

%C 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.

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

%H Amiram Eldar, <a href="/A321869/b321869.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21137.pdf">On some problems of W. Sierpinski</a>, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's Hypothesis H</a>.

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

%t Select[Range[1000], PrimeQ[3#^2 + 2# + 10] && PrimeQ[9#^2 + 6# + 28]  &]

%o (PARI) isok(n) = isprime(m=3*n^2 + 2*n + 10) && isprime(3*m-2); \\ _Michel Marcus_, Nov 20 2018

%Y Cf. A000567, A001567, A321868.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 20 2018