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%I #29 Aug 12 2019 14:00:14
%S 23,29,41,53,89,113,131,179,191,233,239,251,281,293,341,359,419,431,
%T 443,491,509,593,641,653,659,683,719,743,761,809,911,953,1013,1019,
%U 1031,1049,1103,1223,1229,1271,1289,1409,1439,1451,1481,1499,1511,1559,1583,1601,1733,1811,1889,1901,1931,1973,2003
%N 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).
%C 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:
%C (i) primes n such that 2n+1 is prime (cf. A005384) and A007583(n) is composite, with smallest such term n=a(1)=23;
%C (ii) primes n==2 (mod 3) such that 2n+1 is a 2-psp (no such terms are known);
%C (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;
%C (iv) 4-pseudoprimes n==5 (mod 6) such that 2n+1 is 2-pseudoprime, with smallest such term n=268435455;
%C (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;
%C (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.
%Y Cf. A002450, A007583, A175625, A175942, A300193, A303008, A303447, A303448.
%K nonn
%O 1,1
%A _Max Alekseyev_, Apr 23 2018
%E Edited by _Max Alekseyev_, Aug 08 2019
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