OFFSET
1,2
COMMENTS
With 1, exponents of A141453 (see comment by Wolfdieter Lang, Mar 28 2012).
Numbers m such that (1 + k)^m + (-k)^m is prime:
0 (k = 0);
this sequence (k = 1);
A283653 (k = 2);
0, 3, 4, 7, 16, 17, ... (k = 3);
0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);
0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ... (k = 5);
0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);
0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);
0, 4, 7, 29, 31, 32, 67, ... (k = 8);
0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);
0, 3, 5, 19, 32, ... (k = 10);
0, 3, 7, 89, 101, ... (k = 11);
0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);
0, 3, 4, 11, 83, ... (k = 13);
0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);
0, 43, ... (k = 15);
0, 4, 5, 7, 79, ... (k = 16);
0, 2, 3, 8, 13, 71, ... (k = 17);
0, 1607, ... (k = 18);
...
Numbers m such that (1 + k)^m + (-k)^m is not an odd prime for k <= m: 0, 1, 15, 18, 53, 59, 106, 114, 124, 132, 133, 143, 177, 214, 232, 234, 240, 256, ...
Conjecture: if (1 + y)^x + (-y)^x is a prime number then x is zero, or an even power of two, or an odd prime number.
The above conjecture can be proved by considering algebraic factorizations of the polynomials involved. - Jeppe Stig Nielsen, Feb 19 2023
Appears to be essentially the same as A174269. - R. J. Mathar, May 21 2017
LINKS
Jeppe Stig Nielsen, Table of n, a(n) for n = 1..52
FORMULA
a(n) = A174269(n) for n > 2. - Jeppe Stig Nielsen, Feb 19 2023
EXAMPLE
4 is in this sequence because 2^4 + (-1)^4 = 17 is prime.
5 is in this sequence because 2^5 + (-1)^5 = 31 is prime.
MATHEMATICA
Select[Range[0, 10^4], PrimeQ[2^# + (-1)^#] &] (* Michael De Vlieger, May 03 2017 *)
PROG
(Magma) [m: m in [0..1000]| IsPrime(2^m + (-1)^m)];
(PARI) is(m)=ispseudoprime(2^m+(-1)^m) \\ Charles R Greathouse IV, Jun 06 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Juri-Stepan Gerasimov, Apr 28 2017
STATUS
approved