cp's OEIS Frontend

A255859 Least m > 0 such that gcd(m^n+9,(m+1)^n+9) > 1, or 0 if there is no such m.

%I A255859 #26 Aug 28 2020 00:13:13
%S A255859 1,0,18,533,1,32,288,484,1,364,6,176427,1,31239,533,8,1,
%T A255859 8424432925592889329288197322308900672459420460792433,30,16561,1,4,6,
%U A255859 349,1,32,546,2579,1,375766,11,5061867704425915,1,5620,6,8,1
%N A255859 Least m > 0 such that gcd(m^n+9,(m+1)^n+9) > 1, or 0 if there is no such m.
%C A255859 See A118119, which is the main entry for this class of sequences.
%H A255859 Will Wei, <a href="https://www.youtube.com/watch?v=L4ArlAfKTLA">Patterns that appear to hold, but don't - 8424432925592889329288197322308900672459420460792433</a>, video (2020)
%F A255859 a(4k)=1 for k>=0, because gcd(1^(4k)+9, 2^(4k)+9) = gcd(10, 16^k-1) = 5.
%e A255859 For n=1, gcd(m^n+9, (m+1)^n+9) = gcd(m+9, m+10) = 1, therefore a(1)=0.
%e A255859 For n=2, we have gcd(18^2+9, 19^2+9) = gcd(333, 370) = 37, and the pair (m,m+1)=(18,19) is the smallest which yields a GCD > 1, therefore a(2)=37.
%e A255859 For n=4k, see formula.
%t A255859 A255859[n_] := Module[{m = 1}, While[GCD[m^n + 9, (m + 1)^n + 9] <= 1, m++]; m]; Join[{1, 0}, Table[A255859[n], {n, 2, 16}]] (* _Robert Price_, Oct 16 2018 *)
%o A255859 (PARI) a(n,c=9,L=10^7,S=1)={n!=1&&for(a=S,L,gcd(a^n+c,(a+1)^n+c)>1&&return(a))}
%Y A255859 Cf. A118119, A255832, A255852-A255869
%K A255859 nonn,hard,more
%O A255859 0,3
%A A255859 _M. F. Hasler_, Mar 08 2015
%E A255859 a(17)-a(30) from _Hiroaki Yamanouchi_, Mar 12 2015
%E A255859 a(31)-a(36) from _Max Alekseyev_, Aug 06 2015