OFFSET
0,3
COMMENTS
See A118119, which is the main entry for this class of sequences.
a(43) <= 291613846670877. - Max Alekseyev, Aug 07 2015
FORMULA
For k>=0, a(6k+4)=3 because gcd(3^(6k+4)+3, 4^(6k+4)+3) = gcd(9^(3k+2)+3, 16^(3k+2)+3) and 9 = 16 = 2 (mod 7) and 2^(3k+2)+3 = 2^2+3 = 0 (mod 7), so the GCD is a positive multiple of 7.
EXAMPLE
For n=1, gcd(k^n+3, (k+1)^n+3) = gcd(k+3, k+4) = 1, therefore a(1)=0.
For n=2, we have gcd(6^2+3, 7^2+3) = gcd(39, 52) = 13, and the pair (k,k+1)=(6,7) is the smallest which yields a GCD > 1, therefore a(2)=6.
MATHEMATICA
A255853[n_] := Module[{m = 1}, While[GCD[m^n + 3, (m + 1)^n + 3] <= 1, m++]; m]; Join[{1, 0}, Table[A255853[n], {n, 2, 10}]] (* Robert Price, Oct 15 2018 *)
PROG
(PARI) a(n, c=3, 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 A255853(n):
if n == 0: return 1
k = 0
for p in primefactors(resultant(x**n+3, (x+1)**n+3)):
for d in (a for a in sorted(nthroot_mod(-3, n, p, all_roots=True)) if pow(a+1, n, p)==-3%p):
k = min(d, k) if k else d
break
return int(k) # Chai Wah Wu, May 08 2024
CROSSREFS
KEYWORD
nonn,hard
AUTHOR
M. F. Hasler, Mar 08 2015
EXTENSIONS
a(11)-a(40) from Hiroaki Yamanouchi, Mar 12 2015
a(41)-a(42) from Max Alekseyev, Aug 06 2015
STATUS
approved