A285929 - OEIS

%I #44 Mar 25 2023 00:06:43

%S 0,2,3,4,5,7,8,13,16,17,19,31,61,89,107,127,521,607,1279,2203,2281,

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

%U 132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,20996011,24036583,25964951,30402457,32582657,37156667

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

%C With 1, exponents of A141453 (see comment by _Wolfdieter Lang_, Mar 28 2012).

%C Numbers m such that (1 + k)^m + (-k)^m is prime:

%C 0 (k = 0);

%C this sequence (k = 1);

%C A283653 (k = 2);

%C 0, 3, 4, 7, 16, 17, ... (k = 3);

%C 0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);

%C 0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ... (k = 5);

%C 0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);

%C 0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);

%C 0, 4, 7, 29, 31, 32, 67, ... (k = 8);

%C 0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);

%C 0, 3, 5, 19, 32, ... (k = 10);

%C 0, 3, 7, 89, 101, ... (k = 11);

%C 0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);

%C 0, 3, 4, 11, 83, ... (k = 13);

%C 0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);

%C 0, 43, ... (k = 15);

%C 0, 4, 5, 7, 79, ... (k = 16);

%C 0, 2, 3, 8, 13, 71, ... (k = 17);

%C 0, 1607, ... (k = 18);

%C ...

%C 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, ...

%C 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.

%C The above conjecture can be proved by considering algebraic factorizations of the polynomials involved. - _Jeppe Stig Nielsen_, Feb 19 2023

%C Appears to be essentially the same as A174269. - _R. J. Mathar_, May 21 2017

%H Jeppe Stig Nielsen, <a href="/A285929/b285929.txt">Table of n, a(n) for n = 1..52</a>

%F a(n) = A174269(n) for n > 2. - _Jeppe Stig Nielsen_, Feb 19 2023

%e 4 is in this sequence because 2^4 + (-1)^4 = 17 is prime.

%e 5 is in this sequence because 2^5 + (-1)^5 = 31 is prime.

%t Select[Range[0, 10^4], PrimeQ[2^# + (-1)^#] &] (* _Michael De Vlieger_, May 03 2017 *)

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

%o (PARI) is(m)=ispseudoprime(2^m+(-1)^m) \\ _Charles R Greathouse IV_, Jun 06 2017

%Y Supersequence of A000043.

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

%K nonn

%O 1,2

%A _Juri-Stepan Gerasimov_, Apr 28 2017