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A320599 Numbers k such that 4k + 1 and 8k + 1 are both primes. 2

%I #86 Sep 24 2021 07:14:03

%S 9,24,39,57,84,144,150,165,207,219,234,249,252,267,309,324,357,402,

%T 414,507,522,534,555,570,639,654,759,765,777,792,795,882,924,927,942,

%U 969,1044,1065,1089,1155,1200,1215,1227,1389,1395,1437,1509,1530,1554,1557

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

%C 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.

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

%H Seiichi Manyama, <a href="/A320599/b320599.txt">Table of n, a(n) for n = 1..10000</a>

%H Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21137.pdf">On some problems of W. Sierpinski</a>, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's Hypothesis H</a>.

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

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

%o (PARI) isok(n) = isprime(4*n+1) && isprime(8*n+1); \\ _Michel Marcus_, Nov 20 2018

%o (Python)

%o from sympy import isprime

%o def ok(n): return isprime(4*n + 1) and isprime(8*n + 1)

%o print(list(filter(ok, range(1558)))) # _Michael S. Branicky_, Sep 24 2021

%Y Cf. A000217, A001567, A293622.

%Y Intersection of A005098 and A005123.

%K nonn

%O 1,1

%A _Amiram Eldar_, Nov 20 2018

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)