%I #22 May 31 2024 22:10:05

%S 3,5,7,8,9,10,15,17,24,26,28,31,33,35,37,48,50,63,65,80,82,99,101,120,

%T 122,124,126,127,129,143,145,168,170,195,197,215,217,224,226,242,244,

%U 255,257,288,290,323,325,342,344,360,362,399,401,440,442,483,485,511

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

%C Named after the British mathematician Allan Joseph Champneys Cunningham (1842-1928). - _Amiram Eldar_, Apr 02 2022

%H Amiram Eldar, <a href="/A080262/b080262.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CunninghamNumber.html">Cunningham Number</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cunningham_number">Cunningham number</a>.

%F 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

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

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

%Y Cf. A001597 (the perfect powers).

%K nonn

%O 1,1

%A _David W. Wilson_, Feb 11 2003