A285929 Numbers m such that 2^m + (-1)^m is prime.

0, 2, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 28 2017

Keywords

• nonn

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

Examples

			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.
		

Links

• Jeppe Stig Nielsen, Table of n, a(n) for n = 1..52

Crossrefs

Supersequence of A000043.

Cf. A000668, A019434, A141453, A174269, A283653, A285886, A285887, A285888.

Programs

Magma

[m: m in [0..1000]| IsPrime(2^m + (-1)^m)];

Mathematica

Select[Range[0, 10^4], PrimeQ[2^# + (-1)^#] &] (* Michael De Vlieger, May 03 2017 *)

PARI

is(m)=ispseudoprime(2^m+(-1)^m) \\ Charles R Greathouse IV, Jun 06 2017

Formula

a(n) = A174269(n) for n > 2. - Jeppe Stig Nielsen, Feb 19 2023