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

%I A255864 #24 Oct 16 2018 17:26:02
%S A255864 1,0,1,12,1,15,1,2,1,1929501,1,13228907223310811104028677,1,94,1,11,1,
%T A255864 85364353,1,1563,1,49,1,9258095644888888790279763522646107297983,1,23,
%U A255864 1,66,1
%N A255864 Least m > 0 such that gcd(m^n+14, (m+1)^n+14) > 1, or 0 if there is no such m.
%C A255864 See A118119, which is the main entry for this class of sequences.
%C A255864 a(29) with 141 decimal digits is too large to include here (see b-file).
%H A255864 Max Alekseyev, <a href="/A255864/b255864.txt">Table of n, a(n) for n = 0..40</a>
%F A255864 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).
%e A255864 For n=1, gcd(m^n+14, (m+1)^n+14) = gcd(m+14, m+15) = 1, therefore a(1)=0.
%e A255864 For n=0 and n=2, see formula with k=0 and k=1.
%e A255864 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.
%t A255864 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 *)
%o A255864 (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))}
%Y A255864 Cf. A118119, A255832, A255852-A255869
%K A255864 nonn,hard
%O A255864 0,4
%A A255864 _M. F. Hasler_, Mar 09 2015
%E A255864 a(11)-a(40) from _Max Alekseyev_, Aug 06 2015