A255861 Least m > 0 such that gcd(m^n+11, (m+1)^n+11) > 1, or 0 if there is no such m.

1, 0, 1, 2, 1, 23, 1, 19010820161, 1, 7, 1, 360, 1, 41953103, 1, 4, 1, 638386957517954762853, 1, 38884, 1, 2, 1, 2852, 1, 23, 1, 102, 1, 8384, 1, 36556, 1, 33, 1, 37, 1, 336, 1, 2, 1, 1123, 1, 19734, 1, 9, 1, 135356, 1, 399351, 1, 33, 1

Offset

0,4

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(2k)=1 for k>=0, because gcd(1^(2k)+11, 2^(2k)+11) = gcd(12, 4^k-1) = 3.

Example

For n=1, gcd(m^n+11, (m+1)^n+11) = gcd(m+11, m+12) = 1, therefore a(1)=0.
For n=2, we have gcd(2^2+11, 3^2+11) = gcd(15, 20) = 5, and the pair (m,m+1)=(2,3) is the smallest which yields a GCD > 1, therefore a(2)=2.

Mathematica

A255861[n_] := Module[{m = 1}, While[GCD[m^n + 11, (m + 1)^n + 11] <= 1, m++]; m]; Join[{1, 0}, Table[A255861[n], {n, 2, 6}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=11, 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: A330354 A200859 A127607 * A059360 A368016 A279308

Adjacent sequences: A255858 A255859 A255860 * A255862 A255863 A255864

Keyword

nonn

Author

M. F. Hasler, Mar 08 2015

Extensions

a(7)-a(48) from Hiroaki Yamanouchi, Mar 12 2015

a(49)-a(52) from Max Alekseyev, Aug 06 2015

Status

approved