A080262 Cunningham numbers: of the form a^b +- 1, where a, b >= 2.

3, 5, 7, 8, 9, 10, 15, 17, 24, 26, 28, 31, 33, 35, 37, 48, 50, 63, 65, 80, 82, 99, 101, 120, 122, 124, 126, 127, 129, 143, 145, 168, 170, 195, 197, 215, 217, 224, 226, 242, 244, 255, 257, 288, 290, 323, 325, 342, 344, 360, 362, 399, 401, 440, 442, 483, 485, 511

Offset

1,1

Comments

Named after the British mathematician Allan Joseph Champneys Cunningham (1842-1928). - Amiram Eldar, Apr 02 2022

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Cunningham Number.

Wikipedia, Cunningham number.

Formula

a(2n) = A001597(n+2)-1, a(2n+1) = A001597(n+2)+1 for n >= 5, if (25,27) is the only pair of perfect powers that differ by 2. (Note that it is known as Mihăilescu's theorem (formerly called Catalan's conjecture) that (8,9) is the only pair of perfect powers who differ by 1.) - Jianing Song, Oct 15 2022

Example

26 = 3^3 - 1, 126 = 5^3 + 1 are Cunningham numbers.

Mathematica

powerQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[2^9], powerQ[# - 1] || powerQ[# + 1] &] (* Amiram Eldar, Jul 27 2019 *)

Crossrefs

Cf. A001597 (the perfect powers).

Sequence in context: A111638 A324334 A370600 * A025050 A196115 A336867

Adjacent sequences: A080259 A080260 A080261 * A080263 A080264 A080265

Keyword

nonn

Author

David W. Wilson, Feb 11 2003

Status

approved