login
A255852
Least k > 0 such that gcd(k^n+2, (k+1)^n+2) > 1, or 0 if there is no such k.
21
1, 0, 1, 51, 1, 40333, 1, 434, 1, 16, 1, 1234, 1, 78607, 1, 8310, 1, 817172, 1, 473, 1, 116, 1, 22650, 1, 736546059, 1, 22, 1, 1080982, 1, 252, 1, 7809, 1, 644, 1, 1786225573, 1
OFFSET
0,4
COMMENTS
See A118119, which is the main entry for this class of sequences.
a(39) <= 8105110304875691067. - Max Alekseyev, Aug 06 2015
a(41) = 34290868, a(49) <= 2002111070, a(47) = 32286649814088452353414982038778088771611290478685407234712300075870593693164721\
99455164873287615636327176797646292254029648497024652505965417768073756378034012\
80883965289152013363422286845290874810700297549641281106223286199677401563701715\
56997846264124867393209579875386439424082082891813462700417531719383529314983727. - Hiroaki Yamanouchi, Mar 10 2015
a(43) = 3585, a(45) = 5, a(51) = 16, a(57) = 22, a(59) = 4495, a(63) = 1291, a(65) = 108, a(67) = 220, a(69) = 218039, a(71) = 2112. - Chai Wah Wu, May 08 2024
FORMULA
a(2k)=1 for k>=0, because gcd(1^(2k)+2,2^(2k)+2) = gcd(3,4^k-1) = 3.
a(2k+1) = A255832(k).
EXAMPLE
For n=1, gcd(k^n+2,(k+1)^n+2) = gcd(k+2,k+3) = 1, therefore a(1)=0.
For n=2k, see formula.
For n=3, we have gcd(51^3+2,52^3+2) = 109, and the pair (k,k+1)=(51,52) is the smallest which yields a GCD > 1, therefore a(3)=51.
MATHEMATICA
A255852[n_] := Module[{m = 1}, While[GCD[m^n + 2, (m + 1)^n + 2] <= 1, m++]; m];
Join[{1, 0}, Table[A255852[n], {n, 2, 24}]]
PROG
(PARI) a(n, c=2, 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 A255852(n):
if n == 0: return 1
k = 0
for p in primefactors(resultant(x**n+2, (x+1)**n+2)):
for d in (a for a in sorted(nthroot_mod(-2, n, p, all_roots=True)) if pow(a+1, n, p)==-2%p):
k = min(d, k) if k else d
break
return k # Chai Wah Wu, May 08 2024
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
M. F. Hasler, Mar 08 2015
EXTENSIONS
a(25),a(37),a(41),a(47) conjectured by Hiroaki Yamanouchi, Mar 10 2015; confirmed by Max Alekseyev, Aug 06 2015
STATUS
approved