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

%I A285929 #44 Mar 25 2023 00:06:43
%S A285929 0,2,3,4,5,7,8,13,16,17,19,31,61,89,107,127,521,607,1279,2203,2281,
%T A285929 3217,4253,4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,
%U A285929 132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,20996011,24036583,25964951,30402457,32582657,37156667
%N A285929 Numbers m such that 2^m + (-1)^m is prime.
%C A285929 With 1, exponents of A141453 (see comment by _Wolfdieter Lang_, Mar 28 2012).
%C A285929 Numbers m such that (1 + k)^m + (-k)^m is prime:
%C A285929 0 (k = 0);
%C A285929 this sequence (k = 1);
%C A285929 A283653 (k = 2);
%C A285929 0, 3, 4, 7, 16, 17, ... (k = 3);
%C A285929 0, 2, 3, 4, 43, 59, 191, 223, ... (k = 4);
%C A285929 0, 2, 5, 8, 11, 13, 16, 23, 61, 83, ... (k = 5);
%C A285929 0, 3, 4, 7, 16, 29, 41, 67, ... (k = 6);
%C A285929 0, 2, 7, 11, 16, 17, 29, 31, 79, 43, 131, 139, ... (k = 7);
%C A285929 0, 4, 7, 29, 31, 32, 67, ... (k = 8);
%C A285929 0, 2, 3, 4, 7, 11, 19, 29, ... (k = 9);
%C A285929 0, 3, 5, 19, 32, ... (k = 10);
%C A285929 0, 3, 7, 89, 101, ... (k = 11);
%C A285929 0, 2, 4, 17, 31, 32, 41, 47, 109, 163, ... (k = 12);
%C A285929 0, 3, 4, 11, 83, ... (k = 13);
%C A285929 0, 2, 3, 4, 16, 43, 173, 193, ... (k = 14);
%C A285929 0, 43, ... (k = 15);
%C A285929 0, 4, 5, 7, 79, ... (k = 16);
%C A285929 0, 2, 3, 8, 13, 71, ... (k = 17);
%C A285929 0, 1607, ... (k = 18);
%C A285929 ...
%C A285929 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 A285929 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 A285929 The above conjecture can be proved by considering algebraic factorizations of the polynomials involved. - _Jeppe Stig Nielsen_, Feb 19 2023
%C A285929 Appears to be essentially the same as A174269. - _R. J. Mathar_, May 21 2017
%H A285929 Jeppe Stig Nielsen, <a href="/A285929/b285929.txt">Table of n, a(n) for n = 1..52</a>
%F A285929 a(n) = A174269(n) for n > 2. - _Jeppe Stig Nielsen_, Feb 19 2023
%e A285929 4 is in this sequence because 2^4 + (-1)^4 = 17 is prime.
%e A285929 5 is in this sequence because 2^5 + (-1)^5 = 31 is prime.
%t A285929 Select[Range[0, 10^4], PrimeQ[2^# + (-1)^#] &] (* _Michael De Vlieger_, May 03 2017 *)
%o A285929 (Magma) [m: m in [0..1000]| IsPrime(2^m + (-1)^m)];
%o A285929 (PARI) is(m)=ispseudoprime(2^m+(-1)^m) \\ _Charles R Greathouse IV_, Jun 06 2017
%Y A285929 Supersequence of A000043.
%Y A285929 Cf. A000668, A019434, A141453, A174269, A283653, A285886, A285887, A285888.
%K A285929 nonn
%O A285929 1,2
%A A285929 _Juri-Stepan Gerasimov_, Apr 28 2017