A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes.

9, 24, 39, 57, 84, 144, 150, 165, 207, 219, 234, 249, 252, 267, 309, 324, 357, 402, 414, 507, 522, 534, 555, 570, 639, 654, 759, 765, 777, 792, 795, 882, 924, 927, 942, 969, 1044, 1065, 1089, 1155, 1200, 1215, 1227, 1389, 1395, 1437, 1509, 1530, 1554, 1557

Offset

1,1

Comments

Rotkiewicz proved that if k is in this sequence then (4k + 1)*(8k + 1) is a triangular Fermat pseudoprime to base 2 (A293622), and thus under Schinzel's Hypothesis H there are infinitely many triangular Fermat pseudoprimes to base 2.

The corresponding pseudoprimes are 2701, 18721, 49141, 104653, 226801, 665281, 721801, ...

Links

Seiichi Manyama, 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

9 is in the sequence since 4*9 + 1 = 37 and 8*9 + 1 = 73 are both primes.

Mathematica

Select[Range[1000], PrimeQ[4#+1] && PrimeQ[8#+1] &]

Prog

(PARI) isok(n) = isprime(4*n+1) && isprime(8*n+1); \\ Michel Marcus, Nov 20 2018
(Python)
from sympy import isprime
def ok(n): return isprime(4*n + 1) and isprime(8*n + 1)
print(list(filter(ok, range(1558)))) # Michael S. Branicky, Sep 24 2021

Crossrefs

Cf. A000217, A001567, A293622.

Intersection of A005098 and A005123.

Sequence in context: A003343 A345843 A047720 * A293846 A212462 A161449

Adjacent sequences: A320596 A320597 A320598 * A320600 A320601 A320602

Keyword

nonn

Author

Amiram Eldar, Nov 20 2018

Status

approved