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A255863 Least m > 0 such that gcd(m^n+13, (m+1)^n+13) > 1, or 0 if there is no such m. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)