A255863 Least m > 0 such that gcd(m^n+13, (m+1)^n+13) > 1, or 0 if there is no such m.

1, 0, 26, 1, 5, 24308100, 1, 329, 71, 1, 6, 59, 1, 135, 5, 1, 23, 7711, 1, 82, 6, 1, 8, 320594291825643656342, 1, 45, 10, 1, 755, 1107, 1, 4279, 30269, 1, 5, 205961, 1, 259, 8, 1, 9, 101975, 1, 6491, 5, 1, 8

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..46.

Formula

a(3k) = 1 for k>=0, because 1^(3k)+13 = 14, 2^(3k)+13 = 8^k+13 = 14 (mod 7), therefore gcd(1^(3k)+13, 2^(3k)+13) >= 7.

Example

For n=1, gcd(m^n+13, (m+1)^n+13) = gcd(m+13, m+14) = 1, therefore a(1)=0.
For n=2, gcd(26^2+13, 27^2+13) = 53, and (m, m+1) = (26, 27) is the smallest pair which yields a GCD > 1 here.
For n=0, n=3, n=6,... see formula.

Mathematica

A255863[n_] := Module[{m = 1}, While[GCD[m^n + 13, (m + 1)^n + 13] <= 1, m++]; m]; Join[{1, 0}, Table[A255863[n], {n, 2, 22}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=13, 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: A040693 A040692 A040694 * A040695 A040696 A040697

Adjacent sequences: A255860 A255861 A255862 * A255864 A255865 A255866

Keyword

nonn,hard

Author

M. F. Hasler, Mar 10 2015

Extensions

a(5)-a(46) from Hiroaki Yamanouchi, Mar 12 2015

Status

approved