A303009 Numbers n such that both A002450(n)=(2^(2n)-1)/3 and A007583(n)=2*A002450(n)+1 are Fermat pseudoprimes to base 2 (A001567).

23, 29, 41, 53, 89, 113, 131, 179, 191, 233, 239, 251, 281, 293, 341, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1271, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003

Offset: 1

Author

Max Alekseyev, Apr 23 2018

Keywords

• nonn

Comments

It can be shown that if n is odd, it is a prime or a Fermat 4-pseudoprime (A020136) not divisible by 3. Similarly, 2n+1 is a prime or a Fermat 2-pseudoprime (A001567) not divisible by 3. In fact, the sequence is the union of the following six:

(i) primes n such that 2n+1 is prime (cf. A005384) and A007583(n) is composite, with smallest such term n=a(1)=23;

(ii) primes n==2 (mod 3) such that 2n+1 is a 2-psp (no such terms are known);

(iii) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is prime and A007583(n) is composite, with smallest such term n=a(15)=341;

(iv) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is 2-pseudoprime, with smallest such term n=268435455;

(v) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is prime and A007583(n) is composite, with the smallest such term n=67166;

(vi) n=2k, where 4k is in A015921 and k==1 (mod 3), such that 2n+1 is a 2-psp, with the smallest such term n=9042986.

Crossrefs

Cf. A002450, A007583, A175625, A175942, A300193, A303008, A303447, A303448.

Extensions

Edited by Max Alekseyev, Aug 08 2019