A255862 Least m > 0 such that gcd(m^n+12, (m+1)^n+12) > 1, or 0 if there is no such m.

1, 0, 3, 1926, 96, 6, 2, 26, 3, 320, 538, 27, 1, 145, 3, 6, 393216, 982, 3, 2557, 3, 2, 30, 18781248, 1, 6, 3, 188, 14, 145, 3, 2808, 3, 16, 24340653915, 6, 1

Offset: 0

Author

M. F. Hasler, Mar 09 2015

Keywords

• nonn
• hard
• more

Comments

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

Examples

			For n=0, gcd(m^0+12, (m+1)^0+12) = gcd(13, 13) = 13, therefore a(1)=1, the smallest possible (positive) m-value.
For n=1, gcd(m^n+12, (m+1)^n+12) = gcd(m+12, m+13) = 1, therefore a(1)=0.
For n=2, see formula with k=0.
		

Crossrefs

Cf. A118119, A255832, A255852-A255869

Programs

Mathematica

A255862[n_] := Module[{m = 1}, While[GCD[m^n + 12, (m + 1)^n + 12] <= 1, m++]; m]; Join[{1, 0}, Table[A255862[n], {n, 2, 22}]] (* Robert Price, Oct 16 2018 *)

PARI

a(n,c=12,L=10^7,S=1)={n!=1 && for(a=S,L,gcd(a^n+c,(a+1)^n+c)>1 && return(a))}

Formula

a(6k+2) = 3 for k>=0, because 3^(6k+2) = 9^(3k+1), 4^(6k+2) = 16^(3k+1), and 9 = 16 = 2 (mod 7), 2^3 = 1 (mod 7) and 12 = -2 (mod 7), therefore 3^(6k+2)+12 = 4^(6k+2)+12 = 0 (mod 7) and gcd(3^(6k+2)+12, 4^(6k+2)+12) >= 7.

Extensions

a(23)-a(33) from Hiroaki Yamanouchi, Mar 13 2015

a(34)-a(36) from Max Alekseyev, Aug 07 2015