A255857 Least k > 0 such that gcd(k^n+7,(k+1)^n+7) > 1, or 0 if there is no such k.

1, 0, 14, 320, 56, 675113, 224, 13, 5, 283, 33, 192, 26, 242, 5, 2, 10, 140, 5, 50, 142, 29, 18, 605962, 11, 97, 234881024, 951, 5, 3332537854, 14

Offset: 0

Author

M. F. Hasler, Mar 08 2015

Keywords

• nonn

Comments

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

Examples

			For n=0, gcd(k^0+7, (k+1)^0+7) = gcd(8, 8) = 8 for any k > 0, therefore a(0)=1 is the smallest possible positive value.
For n=1, gcd(k^n+7, (k+1)^n+7) = gcd(k+7, k+8) = 1, therefore a(1)=0.
For n=2, we have gcd(14^2+7, 15^2+7) = gcd(203, 232) = 29, and the pair (k,k+1)=(14,15) is the smallest which yields a gcd > 1, therefore a(2)=14.
		

Links

• Hiroaki Yamanouchi, Table of n, a(n) for n = 0..36

Crossrefs

Cf. A118119, A255832, A255852-A255869

Programs

Mathematica

A255857[n_] := Module[{m = 1}, While[GCD[m^n + 7, (m + 1)^n + 7] <= 1, m++]; m]; Join[{1, 0}, Table[A255857[n], {n, 2, 25}]] (* Robert Price, Oct 15 2018 *)

PARI

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

Python

from sympy import primefactors, resultant, nthroot_mod
from sympy.abc import x
def A255857(n):
    if n == 0: return 1
    k = 0
    for p in primefactors(resultant(x**n+7,(x+1)**n+7)):
        for d in (a for a in sorted(nthroot_mod(-7,n,p,all_roots=True)) if pow(a+1,n,p)==-7%p):
            k = min(d,k) if k else d
            break
    return int(k) # Chai Wah Wu, May 09 2024

Extensions

a(26)-a(36) from Hiroaki Yamanouchi, Mar 12 2015