A255864 Least m > 0 such that gcd(m^n+14, (m+1)^n+14) > 1, or 0 if there is no such m.

1, 0, 1, 12, 1, 15, 1, 2, 1, 1929501, 1, 13228907223310811104028677, 1, 94, 1, 11, 1, 85364353, 1, 1563, 1, 49, 1, 9258095644888888790279763522646107297983, 1, 23, 1, 66, 1

Offset

0,4

Comments

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

a(29) with 141 decimal digits is too large to include here (see b-file).

Links

Max Alekseyev, Table of n, a(n) for n = 0..40

Formula

a(2k) = 1 for k>=0, because gcd(1^(2k)+14, 2^(2k)+14) = gcd(15, 4^k-1) >= 3, since 4^k-1 = 1-1 = 0 (mod 3).

Example

For n=1, gcd(m^n+14, (m+1)^n+14) = gcd(m+14, m+15) = 1, therefore a(1)=0.
For n=0 and n=2, see formula with k=0 and k=1.
For n=3, gcd(12^3+14, 13^3+14) = 67, and (m, m+1) = (12, 13) is the smallest pair which yields a GCD > 1 here.

Mathematica

A255864[n_] := Module[{m = 1}, While[GCD[m^n + 14, (m + 1)^n + 14] <= 1, m++]; m]; Join[{1, 0}, Table[A255864[n], {n, 2, 10}]] (* Robert Price, Oct 16 2018 *)

Prog

(PARI) a(n, c=14, 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: A059857 A322762 A070649 * A056583 A281823 A139724

Adjacent sequences: A255861 A255862 A255863 * A255865 A255866 A255867

Keyword

nonn,hard

Author

M. F. Hasler, Mar 09 2015

Extensions

a(11)-a(40) from Max Alekseyev, Aug 06 2015

Status

approved