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A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

%I A238739 #55 Sep 08 2022 08:46:07
%S A238739 1,2,6,12,18,30
%N A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.
%C A238739 Intersection of A057732 and A002253. - _Joerg Arndt_, Mar 04 2014
%C A238739 By checking primality of 2^n+3 for values n in A002253, it follows a(7) > 7033641. - _Giovanni Resta_, Mar 08 2014
%C A238739 Exponents of second Fermat prime pairs. - _Juri-Stepan Gerasimov_, Mar 08 2014
%C A238739 From _Juri-Stepan Gerasimov_, Mar 04 2014: (Start)
%C A238739 If prime pair {2^n + (2k+1), (2k+1)*2^n + 1} is called a Fermat prime pair, then numbers n such that 2^n + (2k + 1) and (2k + 1)*2^n + 1 are both prime:
%C A238739 for k = 0: 0, 1, 2, 4, 8, 16, ...   the exponents first Fermat prime pairs;
%C A238739 for k = 1: 1, 2, 6, 12, 18, 30, ... the exponents second Fermat prime pairs;
%C A238739 for k = 2: 1, 3, ...                the exponents third Fermat prime pairs;
%C A238739 for k = 3: 2, 4, 6, 20, 174, ...    the exponents fourth Fermat prime pairs;
%C A238739 for k = 4: 1, 2, 3, 6, 7, ...       the exponents fifth Fermat prime pairs;
%C A238739 for k = 5: 1, 3, 5, 7, ...          the exponents sixth Fermat prime pairs;
%C A238739 for k = 6: 2, 8, 20, ...            the exponents seventh Fermat prime pairs;
%C A238739 for k = 7: 1, 2, 4, 10, 12, ...     the exponents eighth Fermat prime pairs;
%C A238739 for k = 8:
%C A238739 for k = 9: 6, ...                   the exponents tenth Fermat prime pairs;
%C A238739 for k = 10: 1, 4, 5, 7, 16, ...     the exponents eleventh Fermat prime pairs;
%C A238739 for k = 11:
%C A238739 for k = 12: 2, 4, 6, 10, 20, 22, ...the exponents thirteenth Fermat prime pairs;
%C A238739 for k = 13: 2, 4, 16, 40, 44, ...   the exponents fourteenth Fermat prime pairs;
%C A238739 for k = 14: 1, 3, 5, 27, 43, ...    the exponents fifteenth Fermat prime pairs.
%C A238739 Semiprimes of the form (2^m+2k+1)*((2k+1)*2^m+1): 4, 9, 25, 35, 77, 91, 209, 289, 319, 481, 527, 533, 901, 989, ...
%C A238739 (End)
%e A238739 a(1) = 1 because 2^1 + 3 = 5 and 3*2^1 + 1 = 7 are both prime,
%e A238739 a(2) = 2 because 2^2 + 3 = 7 and 3^2^2 + 1 = 13 are both prime,
%e A238739 a(3) = 6 because 2^6 + 3 = 67 and 3*2^6 + 1 = 193 are both prime.
%t A238739 Select[Range[30],AllTrue[{2^#+3,3*2^#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Aug 08 2015 *)
%o A238739 (PARI) isok(n) = isprime(2^n + 3) && isprime(3*2^n + 1); \\ _Michel Marcus_, Mar 04 2014
%o A238739 (Magma) [n: n in [0..30] | IsPrime(2^n+3) and IsPrime(3*2^n+1)]; // _Arkadiusz Wesolowski_, Jan 23 2016
%Y A238739 Cf. A002253, A019434, A039687, A057732, A057733, A238554, A239038, A267984.
%K A238739 nonn,hard,more
%O A238739 1,2
%A A238739 _Juri-Stepan Gerasimov_, Mar 04 2014