A255858 Least k > 0 such that gcd(k^n + 8, (k+1)^n + 8) > 1, or 0 if there is no such k.

1, 0, 1, 5, 1, 5238149, 1, 24747, 1, 5, 1, 1042068640211853141849723, 1, 28913777, 1, 5, 1, 9380, 1, 87940, 1, 5, 1, 677083547, 1, 1584, 1, 5, 1, 5355, 1, 257, 1, 5, 1, 118, 1, 3250, 1, 5, 1, 78609080, 1, 1807714890, 1, 5, 1

Offset: 0

Author

M. F. Hasler, Mar 08 2015

Keywords

• nonn
• hard

Comments

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

Examples

			For n=1, gcd(k^n + 8, (k+1)^n + 8) = gcd(k+8, k+9) = 1, therefore a(1)=0.
For n=2*k, see formula.
For n=3, we have gcd(5^3 + 8, 6^3 + 8) = gcd(133, 224) = 7, and the pair (k,k+1)=(5,6) is the smallest which yields a GCD > 1, therefore a(3)=5.
		

Crossrefs

Cf. A118119, A255832, A255852-A255869.

Programs

Mathematica

A255858[n_] := Module[{m = 1}, While[GCD[m^n + 8, (m + 1)^n + 8] <= 1, m++]; m]; Join[{1, 0}, Table[A255858[n], {n, 2, 10}]] (* Robert Price, Oct 15 2018 *)

PARI

a(n,c=8,L=10^7,S=1)={n!=1&&for(a=S,L,gcd(a^n+c,(a+1)^n+c)>1&&return(a))}

Formula

a(2k)=1 for k >= 0, because gcd(1^(2k) + 8, 2^(2k) + 8) = gcd(9, 4^k-1) >= 3.

Extensions

a(11)-a(40) from Hiroaki Yamanouchi, Mar 13 2015

a(41)-a(46) from Max Alekseyev, Aug 06 2015