A255853 Least k > 0 such that gcd(k^n+3, (k+1)^n+3) > 1, or 0 if there is no such k.

1, 0, 6, 56, 3, 29, 96, 1159823, 384, 9, 3, 1994117680, 13, 247, 6, 15, 3, 1256, 4, 25211925041, 15, 5785, 3, 93602696971, 24, 11, 6, 182, 3, 4644, 92, 12506, 9, 13, 3, 484, 2, 420, 6, 130, 3, 16032496, 12

Offset

0,3

Comments

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

a(43) <= 291613846670877. - Max Alekseyev, Aug 07 2015

Links

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

Formula

For k>=0, a(6k+4)=3 because gcd(3^(6k+4)+3, 4^(6k+4)+3) = gcd(9^(3k+2)+3, 16^(3k+2)+3) and 9 = 16 = 2 (mod 7) and 2^(3k+2)+3 = 2^2+3 = 0 (mod 7), so the GCD is a positive multiple of 7.

Example

For n=1, gcd(k^n+3, (k+1)^n+3) = gcd(k+3, k+4) = 1, therefore a(1)=0.
For n=2, we have gcd(6^2+3, 7^2+3) = gcd(39, 52) = 13, and the pair (k,k+1)=(6,7) is the smallest which yields a GCD > 1, therefore a(2)=6.

Mathematica

A255853[n_] := Module[{m = 1}, While[GCD[m^n + 3, (m + 1)^n + 3] <= 1, m++]; m]; Join[{1, 0}, Table[A255853[n], {n, 2, 10}]] (* Robert Price, Oct 15 2018 *)

Prog

(PARI) a(n, c=3, 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: A132689 A323876 A249672 * A183594 A140790 A335217

Adjacent sequences: A255850 A255851 A255852 * A255854 A255855 A255856

Keyword

nonn,hard

Author

M. F. Hasler, Mar 08 2015

Extensions

a(11)-a(40) from Hiroaki Yamanouchi, Mar 12 2015

a(41)-a(42) from Max Alekseyev, Aug 06 2015

Status

approved