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Numbers k such that exactly one of 2^k - 1 and 2^k + 1 is a prime.
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%I #39 Feb 20 2023 07:52:52

%S 0,1,3,4,5,7,8,13,16,17,19,31,61,89,107,127,521,607,1279,2203,2281,

%T 3217,4253,4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,

%U 132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917

%N Numbers k such that exactly one of 2^k - 1 and 2^k + 1 is a prime.

%C Apart from the first term, all terms are primes (Mersenne exponents) or powers of two (Fermat exponents). The sequence consists of all members of A000043 and A092506, apart from 2. - _Charles R Greathouse IV_, Mar 20 2010

%C Numbers k such that one of 2^k+1 or 2^k-1 is prime, but not both. - _R. J. Mathar_, Mar 29 2010

%C The sequence "Numbers k such that 2^k + (-1)^k is a prime" gives essentially the same sequence, except with the initial 1 replaced by 2. - _Thomas Ordowski_, Dec 26 2016

%C The union of 2 and this sequence gives the values k for which 2^k or 2^k - 1 are the numbers in A006549. - _Gionata Neri_, Dec 19 2015

%C The union of 2 and this sequence is the values k for which either 2^k - 1 or 2^k + 1, or both, are prime. The reason why this only yields one additional term, 2, is because the number 3 always divides either 2^k - 1 or 2^k + 1 (also implicit in Ordowski comment). - _Jeppe Stig Nielsen_, Feb 19 2023

%H Jeppe Stig Nielsen, <a href="/A174269/b174269.txt">Table of n, a(n) for n = 1..52</a>

%F a(n) = A285929(n) for n > 2. - _Jeppe Stig Nielsen_, Feb 19 2023

%e 0 is in the sequence because 2^0 - 1 = 0 is nonprime and 2^0 + 1 = 2 is prime; 2 is not in the sequence because 2^2 - 1 = 3 and 2^2 + 1 = 5 are both prime.

%t Select[Range[0, 5000], Xor[PrimeQ[2^# - 1], PrimeQ[2^# + 1]] &] (* _Michael De Vlieger_, Jan 03 2016 *)

%o (PARI) isok(k) = my(p = 2^k-1, q = p+2); bitxor(isprime(p), isprime(q)); \\ _Michel Marcus_, Jan 03 2016

%Y Cf. A000043, A092506, A019434, A285929.

%K nonn

%O 1,3

%A _Juri-Stepan Gerasimov_, Mar 14 2010

%E a(10)-a(43) from _Charles R Greathouse IV_, Mar 20 2010