A322161 Numbers k such that m = 8k^2 + 2k + 33 and 8m - 7 are both primes.

1, 46, 58, 133, 145, 175, 208, 223, 241, 403, 430, 463, 526, 568, 808, 868, 985, 1015, 1021, 1105, 1120, 1360, 1465, 1501, 1600, 1918, 1978, 2236, 2350, 2413, 2908, 2965, 3043, 3211, 3265, 3523, 3556, 3568, 3601, 3721, 3811, 3868, 4066, 4291, 4300, 4336, 4831

Offset: 1

Author

Amiram Eldar, Nov 29 2018

Keywords

• nonn

Comments

Rotkiewicz proved that if k is in this sequence, and m = 8k^2 + 2k + 33, then m*(8m - 7) is an octadecagonal Fermat pseudoprime to base 2 (A322160), and thus under Schinzel's Hypothesis H there are infinitely many decagonal Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 14491, 2326319101, 5858192341, 160881885091, 227198832571, 481700815831, 960833787841, ...

Examples

			1 is in the sequence since 8*1^2 + 2*1 + 33 = 43 and 8*43 - 7 = 337 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. A001567, A051870, A322160.

Programs

Mathematica

Select[Range[1000], PrimeQ[8#^2 + 2# + 33] && PrimeQ[64#^2 + 16# + 257]  &]

PARI

isok(n) = isprime(m = 8*n^2+2*n+33) && isprime(8*m-7); \\ Michel Marcus, Nov 29 2018