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

Author

David W. Wilson, Feb 11 2003

Keywords

• nonn

Comments

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

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Cunningham Number.
• Wikipedia, Cunningham number.

Crossrefs

Cf. A001597 (the perfect powers).

Programs

Mathematica

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

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