A285886 Primes of the form (1 + x)^y + (-x)^y where x is a divisor of y.

5, 7, 13, 17, 31, 37, 97, 127, 257, 881, 4651, 8191, 65537, 131071, 524287, 1273609, 2147483647, 2305843009213693951, 618970019642690137449562111, 3512911982806776822251393039617, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 27 2017

Keywords

• nonn

Comments

If x = y then: 13, 37, 881, 4651, 1273609, ...

Primes of the form (1 + x)^y - x^y where y is divisor of x: 3, 5, 7, 31, 37, 127, 4651, 8191, 131071, 524287, ..., which is A285887.

Examples

			5 is in this sequence because (1 + 1)^2 + (-1)^2 = 5 is prime where 1 is a divisor of 2.
A complete list of (x, y, p) corresponding to the shown data is
  (1,2,5), (1,3,7), (2,2,13), (1,4,17), (1,5,31), (3,3,37), (2,4,97),(1,7,127), (1,8,257), (4,4,881), (5,5,4651), (1,13,8191), (1,16,65537),
  (1,17,131071), (1,19,524287), (7,7,1273609), (1,31,2147483647),
  (1,61,2305843009213693951), (1,89,618970019642690137449562111),
  (8,32,3512911982806776822251393039617),
  (1,107,162259276829213363391578010288127),
  (1,127,170141183460469231731687303715884105727).
  Further terms correspond to (x,y) = {(1,521), (1,607), (167,167), (1,1279), (1,2203), (1,2281), (1,3217), ...}. - _Hugo Pfoertner_, Jan 12 2020
		

Links

• Georg Fischer, Table of n, a(n) for n = 1..23
• J. S. Gerasimov, x^(y + 1) - y^x, SeqFan list, Aug 18 2014.

Crossrefs

Cf. A000668 (Mersenne primes), A019434 (Fermat primes), A243100, A285887, A285888.

Programs

Mathematica

Union@ Flatten@ Table[Select[Map[(1 + #)^n + (-#)^n &, Divisors@ n], PrimeQ], {n, 150}] (* Michael De Vlieger, Apr 29 2017 *)

Extensions

Edited by N. J. A. Sloane, Jan 11 2020