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A287614 Primes of the form (1 + x)^y + (-x)^y for some positive x, y. 0
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 (list; graph; refs; listen; history; text; internal format)
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) hase 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

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Last modified January 21 13:19 EST 2022. Contains 350479 sequences. (Running on oeis4.)