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

1,2

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, ...

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

1 is in the sequence since 8*1^2 + 2*1 + 33 = 43 and 8*43 - 7 = 337 are both primes.

Mathematica

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

Prog

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

Crossrefs

Cf. A001567, A051870, A322160.

Sequence in context: A332952 A308099 A308252 * A039424 A043247 A044027

Adjacent sequences: A322158 A322159 A322160 * A322162 A322163 A322164

Keyword

nonn

Author

Amiram Eldar, Nov 29 2018

Status

approved