A287614 Primes of the form (1 + x)^y + (-x)^y for some positive x, y.

5, 7, 13, 17, 19, 31, 37, 41, 61, 97, 113, 127, 181, 211, 257, 271, 313, 331, 337, 397, 421, 547, 613, 631, 761, 881, 919, 1013, 1201, 1301, 1657, 1741, 1801, 1861, 1951, 2113, 2269, 2381, 2437, 2521, 2791, 3121, 3169, 3571, 3613, 3697, 4219, 4447, 4513, 4651, 5101, 5167, 5419, 6211

Offset

1,1

Comments

Conjecture: If x is a positive number and (1 + x)^y + (-x)^y is an odd prime number, then y is other odd prime number or even power of two.

Smallest Mersenne prime (A000668) has n ways to write as (1 + k)^m - k^m for positive k: 3, 7, 127, ...

Links

Table of n, a(n) for n=1..54.

Example

5 (x = 1, y = 2), 7 (1, 3), 13 (2, 2), 17 (1, 4), 19 (2, 3), 31 (1, 5), 37 (3, 3), 41 (4, 2), 61 (3, 4 or 2, 5), 97 (2, 4), 113 (7, 2), 127 (1, 7 or 3, 6), 181 (9, 2), 211 (2, 5), 257 (1, 8), 271 (9, 3).

Mathematica

mx = 10^4; f[x_, y_] := (1+x)^y + (-x)^y; x=0; Union@ Reap[ While[ f[++x, 2] < mx, y=1; While[(v = f[x, ++y]) < mx, If[PrimeQ@ v, Sow@v]]]][[2, 1]] (* Giovanni Resta, May 31 2017 *)

Crossrefs

Cf. A000668, A285929, A283653, A286348.

Sequence in context: A091301 A040125 A106067 * A320866 A342705 A314321

Adjacent sequences: A287611 A287612 A287613 * A287615 A287616 A287617

Keyword

nonn

Author

Juri-Stepan Gerasimov, May 27 2017

Status

approved