A285886 - OEIS

%I #38 Dec 23 2024 14:53:45

%S 5,7,13,17,31,37,97,127,257,881,4651,8191,65537,131071,524287,1273609,

%T 2147483647,2305843009213693951,618970019642690137449562111,

%U 3512911982806776822251393039617,162259276829213363391578010288127,170141183460469231731687303715884105727

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

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

%C 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.

%H Georg Fischer, <a href="/A285886/b285886.txt">Table of n, a(n) for n = 1..23</a>

%H J. S. Gerasimov, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-August/013480.html">x^(y + 1) - y^x</a>, SeqFan list, Aug 18 2014.

%e 5 is in this sequence because (1 + 1)^2 + (-1)^2 = 5 is prime where 1 is a divisor of 2.

%e A complete list of (x, y, p) corresponding to the shown data is

%e   (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),

%e   (1,17,131071), (1,19,524287), (7,7,1273609), (1,31,2147483647),

%e   (1,61,2305843009213693951), (1,89,618970019642690137449562111),

%e   (8,32,3512911982806776822251393039617),

%e   (1,107,162259276829213363391578010288127),

%e   (1,127,170141183460469231731687303715884105727).

%e   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

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

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

%K nonn

%O 1,1

%A _Juri-Stepan Gerasimov_, Apr 27 2017

%E Edited by _N. J. A. Sloane_, Jan 11 2020