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A320599
Numbers k such that 4k + 1 and 8k + 1 are both primes.
2
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
Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
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
Intersection of A005098 and A005123.
Sequence in context: A003343 A345843 A047720 * A293846 A212462 A161449
KEYWORD
nonn
AUTHOR
Amiram Eldar, Nov 20 2018
STATUS
approved