A255854 Least k > 0 such that gcd(k^n+4, (k+1)^n+4) > 1, or 0 if there is no such k.

1, 0, 8, 210, 1, 82, 128, 4763358550, 1, 22, 8, 4050643070777669523228, 1, 1010633974733, 7784, 100, 1, 26627469676193276478340, 8, 179, 1, 4082, 48, 1293523748876425462850, 1, 173, 8, 5, 1, 2423, 320, 342, 1, 1162, 8, 93, 1, 455207, 128, 22, 1, 11383, 8, 58768, 1, 91, 96, 306824898, 1, 187751, 8, 84, 1

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..52.

Formula

a(4k)=1, a(8k+2)=8 (k>=0), cf. examples.

Example

For n=1, gcd(k^n+4, (k+1)^n+4) = gcd(k+4, k+5) = 1, therefore a(1)=0.
For n=2, we have gcd(8^2+4, 9^2+4) = gcd(68, 85) = 17, and the pair (k,k+1)=(8,9) is the smallest with this property, therefore a(2)=8.
More generally, a(8k+2)=8 because gcd(8^(8k+2)+4, 9^(8k+2)+4) = gcd(64^(4k+1)+4, 81^(4k+1)+4) >= 17, since 64 = 81 = 13 (mod 17) and 13^4 = 1 (mod 17).
Also a(4k)=1, because gcd(1^(4k)+4, 2^(4k)+4) = gcd(5, 16^k-1) = 5.

Mathematica

A255854[n_] := Module[{m = 1}, While[GCD[m^n + 4, (m + 1)^n + 4] <= 1, m++]; m]; Join[{1, 0}, Table[A255854[n], {n, 2, 6}]] (* Robert Price, Oct 15 2018 *)

Prog

(PARI) a(n, c=4, L=10^6, 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: A279663 A294970 A275286 * A151797 A279464 A197596

Adjacent sequences: A255851 A255852 A255853 * A255855 A255856 A255857

Keyword

nonn

Author

M. F. Hasler, Mar 08 2015

Extensions

a(7)-a(46) from Hiroaki Yamanouchi, Mar 13 2015

a(47)-a(52) from Max Alekseyev, Aug 06 2015

Status

approved