A321871 Numbers k such that m = 4k^2 + 2k + 17 and 4m - 3 are both primes.

1, 3, 4, 9, 11, 14, 23, 36, 38, 49, 66, 101, 133, 134, 141, 154, 158, 191, 193, 196, 198, 213, 228, 241, 269, 283, 294, 313, 334, 339, 384, 394, 411, 413, 431, 453, 499, 511, 554, 558, 601, 619, 639, 649, 661, 686, 701, 704, 718, 758, 791, 804, 818, 821, 881

Offset: 1

Author

Amiram Eldar, Nov 20 2018

Keywords

• nonn

Comments

Rotkiewicz proved that if k is in this sequence, and m = 4k^2 + 2k + 17, then m*(4m - 3) is a decagonal Fermat pseudoprime to base 2 (A321870), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2047, 13747, 31417, 514447, 1092547, 2746477, 18985627, 111202297, 137763037, ...

Examples

			1 is in the sequence since 4*1^2 + 2*1 + 17 = 23 and 4*23 - 3 = 89 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. A001107, A001567, A321870.

Programs

Mathematica

Select[Range[1000], PrimeQ[4#^2 + 2# + 17] && PrimeQ[16#^2 + 8# + 65] &]

PARI

isok(n) = isprime(m=4*n^2 + 2*n + 17) && isprime(4*m-3); \\ Michel Marcus, Nov 20 2018