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

1,2

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, ...

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 4*1^2 + 2*1 + 17 = 23 and 4*23 - 3 = 89 are both primes.

Mathematica

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

Prog

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

Crossrefs

Cf. A001107, A001567, A321870.

Sequence in context: A047459 A177943 A091380 * A050006 A176744 A023420

Adjacent sequences: A321868 A321869 A321870 * A321872 A321873 A321874

Keyword

nonn

Author

Amiram Eldar, Nov 20 2018

Status

approved