A255866 Least m > 0 such that gcd(m^n+16, (m+1)^n+16) > 1, or 0 if there is no such m.

1, 0, 2, 22, 128, 12, 2, 81, 1, 5982, 2, 11417025, 32768, 70471611388086, 2, 26, 1, 1019, 2, 12168420936538713481747, 48, 128, 2, 788, 1, 131711329, 2, 91, 13, 2920553219286322570768516629247, 2, 237, 1, 22, 2, 108, 27, 9404578, 2, 2859801, 1, 41772125, 2

Offset

0,3

Comments

See A118119, which is the main entry for this class of sequences.

Links

Table of n, a(n) for n=0..42.

Formula

a(4k+2) = 2 for k>=0, because 2^(4k+2) = 4^(2k+1), 3^(4k+2) = 9^(2k+1), and 4 = 9 = -1 (mod 5), therefore gcd(2^(4k+2)+16, 3^(4k+2)+16) >= 5.

Example

For n=0, gcd(m^0+16, (m+1)^0+16) = gcd(16, 16) = 16, therefore a(0)=1, the smallest possible (positive) m-value.
For n=1, gcd(m^n+16, (m+1)^n+16) = gcd(m+15, m+16) = 1, therefore a(1)=0.
For n=2, see formula with k=0.
For n=3, gcd(22^3+16, 23^3+16) = 31 and (m, m+1) = (22, 23) is the smallest pair which yields a GCD > 1 here.

Mathematica

A255866[n_] := Module[{m = 1}, While[GCD[m^n + 16, (m + 1)^n + 16] <= 1, m++]; m]; Join[{1, 0}, Table[A255866[n], {n, 2, 10}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=16, L=10^7, S=1)={n!=1 && for(a=S, L, gcd(a^n+c, (a+1)^n+c)>1 && return(a))}

Crossrefs

Cf. A118119, A255832, A255852-A255869

Sequence in context: A292452 A292732 A206418 * A330900 A356341 A353191

Adjacent sequences: A255863 A255864 A255865 * A255867 A255868 A255869

Keyword

nonn,hard

Author

M. F. Hasler, Mar 09 2015

Extensions

a(11)-a(42) from Max Alekseyev, Aug 06 2015

Status

approved