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%I #31 Jun 29 2024 09:09:48
%S 3,4,5,6,7,8,9,11,12,13,14,15,17,18,19,20,21,23,24,25,27,29,30,31,32,
%T 33,35,37,38,39,41,42,43,44,45,47,48,49,51,53,54,55,57,59,60,61,62,63,
%U 65,67,68,69,71,72,73,74,75,77,79,80,81,83,84,85,87,89,90,91,93,95,97,98,99,101
%N Numbers m such that either m^k - 1 or m^k - 2 is prime for some positive k, but not both.
%C From _R. J. Mathar_, Jul 22 2015: (Start)
%C 10 is not in the sequence because 10^k-2 is even and 10^k-1 is divisible by 3 (because 10^k mod 3 = 1 as 10 mod 3 = 1). 16 is not in the sequence because 16^k-2 is even and 16^k-1 is divisible by 3 (because 16^k mod 3 = 1 as 16 mod 3 = 1). For the same reason almost all even numbers of the form 3m+1 (A016957) are absent, the only exception being 4, as 4^1-1 is a prime.
%C 36 is not in the sequence because 36^k-1 is even and 36^k-1 is divisible by 5 (because 36^k mod 5 = 1 as 36 mod 5 = 1). This reasoning excludes all terms of A017341 (except for 6, as 6^1-1 is prime) from this sequence. With the same methology we can fish for (and exclude) even numbers of the form m*p+1 for primes p >= 3. (End)
%e 2 is not in this sequence because 2^2 - 1 = 3 and 2^2 - 2 = 2 are both prime.
%e 3 is in this sequence because 3^1 - 1 = 2 (prime) and 3^1 - 2 = 1 (nonprime) or 3^2 - 1 = 5 (prime) and 3^2 - 2 = 4 (nonprime).
%o (PARI) is(n)=n>2 && if(n%2,1,isprime(n-1)) \\ _Charles R Greathouse IV_, Jun 03 2015
%Y Cf. A016957, A017341.
%K nonn,easy
%O 1,1
%A _Juri-Stepan Gerasimov_, May 23 2015