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A255859 Least m > 0 such that gcd(m^n+9,(m+1)^n+9) > 1, or 0 if there is no such m. 4

%I #26 Aug 28 2020 00:13:13

%S 1,0,18,533,1,32,288,484,1,364,6,176427,1,31239,533,8,1,

%T 8424432925592889329288197322308900672459420460792433,30,16561,1,4,6,

%U 349,1,32,546,2579,1,375766,11,5061867704425915,1,5620,6,8,1

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

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

%H Will Wei, <a href="https://www.youtube.com/watch?v=L4ArlAfKTLA">Patterns that appear to hold, but don't - 8424432925592889329288197322308900672459420460792433</a>, video (2020)

%F a(4k)=1 for k>=0, because gcd(1^(4k)+9, 2^(4k)+9) = gcd(10, 16^k-1) = 5.

%e For n=1, gcd(m^n+9, (m+1)^n+9) = gcd(m+9, m+10) = 1, therefore a(1)=0.

%e 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 For n=4k, see formula.

%t 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 (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 Cf. A118119, A255832, A255852-A255869

%K nonn,hard,more

%O 0,3

%A _M. F. Hasler_, Mar 08 2015

%E a(17)-a(30) from _Hiroaki Yamanouchi_, Mar 12 2015

%E a(31)-a(36) from _Max Alekseyev_, Aug 06 2015

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Last modified March 18 22:29 EDT 2024. Contains 370951 sequences. (Running on oeis4.)