A174269 -id:A174269 - cp's OEIS frontend

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

Juri-Stepan Gerasimov, Apr 28 2017

nonn

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

			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.

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

Supersequence of A000043.

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

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

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

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

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

A264747 Prime powers n such that either n - 1 or n + 1 is a prime power, but not both.

Original entry on oeis.org

1, 5, 7, 9, 16, 17, 31, 32, 127, 128, 256, 257, 8191, 8192, 65536, 65537, 131071, 131072, 524287, 524288, 2147483647, 2147483648, 2305843009213693951, 2305843009213693952, 618970019642690137449562111

Offset: 1

Juri-Stepan Gerasimov, Nov 23 2015

nonn

From Robert Israel, Nov 25 2015: (Start)
By Mihailescu's theorem, the only case where n-1 and n are both in A025475 is n=9. Thus for n > 9 the sequence consists of the following:
     n = 2^p - 1 and 2^p where 2^p-1 is a Mersenne prime (A000668);
     n = 2^(2^m) and 2^(2^m)+1 where 2^(2^m)+1 is a Fermat prime (A019434).
(End)

			7 is in this sequence because 7 and 7 + 1 = 8 are both prime power, but 7 - 1 = 6 is not a prime power.

Robert Israel, Table of n, a(n) for n = 1..42

Cf. A000668, A000961, A019434, A164512, A174269.

fermats:= {seq(2^(2^m)+1, m=1..4)}:
mersennes:= {seq(numtheory:-mersenne([i]), i=2..14)}:
R:= fermats union map(`-`,fermats,1) union mersennes union map(`+`,mersennes,1):
sort(convert(R union {1,9} minus {2,3,4,8},list)); # Robert Israel, Nov 25 2015

is(k) = isprimepower(k) || k==1;
for(k=1, 1e6, if(is(k) && is(k-1) + is(k+1) == 1, print1(k, ", "))) \\ Altug Alkan, Nov 23 2015

A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

Original entry on oeis.org

13, 17, 19, 23, 29, 83, 149, 173, 227, 389, 1109, 4001, 35753, 36551, 363119, 702193

Offset: 1

Juri-Stepan Gerasimov, Jan 18 2020

Primes p such that exactly one of k*2^p - 2*k + 1 and k*2^p + 2*k - 1 is a prime:
k = 1: odd terms in A000043;
k = 2: this sequence;
k = 3: 5, 13, 19, 29, 31, 109, 139, 271, 379, 1553, ...
k = 4: 2, 37, ...
k = 5: 3, 5, 7, 17, 19, 23, 41, 61, 67, 151, 157, 313, 4111, 6337, ...
k = 6: 2, 5, 7, 11, 19, 29, 149, 191, 373, 449, 983, 1667, 1973, ...
k = 7: 2, 3, 5, 7, 11, 13, 29, 43, 61, 97, 109, 127, 131, 239, 461, 1153, ...
k = 8: 3, 11, 19, 23, 29, 37, 43, 97, 193, 307, 617, 1847, ...
k = 9: 3, 5, 23, 41, 61, 71, 97, 131, 157, 863, 3119, ...
k = 10: 2, 3, 13, ...
...

			13 is in this sequence because 2^(13+1) - 3 = 16381 (prime) and 2^(13+1) + 3 = 16387 (composite number).

Cf. A000043, A050414, A057732, A174269.

[p: p in PrimesUpTo(1000) | not (#[k: k in [2] | IsPrime(k*2^p-2*k+1)]) eq (#[k: k in [2] | IsPrime(k*2^p+2*k-1)])];

Select[Range[400], PrimeQ[#] && Xor @@ PrimeQ[2^(# + 1) + {-3, 3}] &] (* Amiram Eldar, Jan 19 2020 *)

isok(p) = isprime(2*2^p-3) + isprime(2*2^p+3) == 1;
forprime(p=2, 500, if(isok(p), print1(p, ", "))); \\ Jinyuan Wang, Jan 19 2020

a(12)-a(16) added using A050414 and A057732 by Jinyuan Wang, May 15 2020

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

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A264747 Prime powers n such that either n - 1 or n + 1 is a prime power, but not both.

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A331487 Primes p such that exactly one of 2^(p+1) - 3 and 2^(p+1) + 3 is a prime.

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Magma

Mathematica

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