A255860 Least m > 0 such that gcd(m^n+10, (m+1)^n+10) > 1, or 0 if there is no such m.

1, 0, 20, 3, 2, 3, 320, 874, 6, 33, 1, 124, 465, 23433448460229, 81920, 3, 2, 82, 65, 2101, 1, 3, 3, 2398892314, 7270, 3, 11, 21, 2, 97546469, 1, 765170730, 6, 15, 3, 3, 23, 370460325141871548, 29206018, 3, 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..40.

Example

For n=0, gcd(m^0+10, (m+1)^0+10) = gcd(11, 11) = 11 for any m > 0, therefore a(0)=1 is the smallest possible positive value.
For n=1, gcd(m^n+10, (m+1)^n+10) = gcd(m+10, m+11) = 1, therefore a(1)=0.
For n=2, we have gcd(20^2+10, 21^2+10) = gcd(410, 451) = 41, and the pair (m,m+1)=(20,21) is the smallest which yields a GCD > 1, therefore a(2)=20.

Mathematica

A255860[n_] := Module[{m = 1}, While[GCD[m^n + 10, (m + 1)^n + 10] <= 1, m++]; m]; Join[{1, 0}, Table[A255860[n], {n, 2, 12}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=10, 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: A040399 A040390 A040391 * A195600 A118295 A070645

Adjacent sequences: A255857 A255858 A255859 * A255861 A255862 A255863

Keyword

nonn,hard

Author

M. F. Hasler, Mar 08 2015

Extensions

a(13)-a(36) from Hiroaki Yamanouchi, Mar 13 2015

a(37)-a(40) from Max Alekseyev, Aug 06 2015

Status

approved