A255867 - OEIS

%I #29 May 08 2024 02:28:22

%S 1,0,1,1925,1,189812175,1,2,1,116,1,55508752881180794569675021,1,

%T 337276,1,230,1,162,1,2628,1,15,1,3604979675443168377172749,1,53,1,

%U 248,1,254,1,5998484614,1,1323,1,2,1,42750021,1,51,1,17870,1,108,1,87,1,8274,1,2,1,35,1,4049,1,308,1,8885,1,2805086,1

%N Least m > 0 such that gcd(m^n+17, (m+1)^n+17) > 1, or 0 if there is no such m.

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

%F a(2k) = 1 for k>=0, because gcd(1^(2k)+17, 2^(2k)+17) = gcd(18, 4^k-1) >= 3 since 4 = 1 (mod 3).

%e For n=0, gcd(m^0+17, (m+1)^0+17) = gcd(18, 18) = 18, therefore a(0)=1, the smallest possible (positive) m-value.

%e For n=1, gcd(m^n+17, (m+1)^n+17) = gcd(m+17, m+18) = 1, therefore a(1)=0.

%e For n=2, see formula with k=0.

%e For n=3, gcd(1925^3+17, 1926^3+17) = 1951 and (m, m+1) = (1925, 1926) is the smallest pair which yields a GCD > 1 here.

%p f:= proc(n) local q1, q2, r, m, bestm,p,A;

%p   q1:= m^n + 17;

%p   q2:= (m+1)^n + 17;

%p   r:= resultant(q1,q2, m);

%p   bestm:= infinity;

%p   for p in numtheory:-factorset(r) do

%p     A:= [msolve(q1, p)];

%p     A:= select(s -> eval(q2, s) mod p = 0, A);

%p     bestm:= min(bestm, op(map(s -> subs(s,m), A)));

%p   od;

%p   if bestm = infinity then -1 else bestm fi

%p end proc:

%p f(0):= 1: f(1):=0:

%p map(f, [$1..26]); # _Robert Israel_, May 31 2019

%t A255867[n_] := Module[{m = 1}, While[GCD[m^n + 17, (m + 1)^n + 17] <= 1, m++]; m]; Join[{1, 0}, Table[A255867[n], {n, 2, 10}]] (* _Robert Price_, Oct 16 2018 *)

%o (PARI) a(n,c=17,L=10^7,S=1)={n!=1 && for(a=S,L,gcd(a^n+c,(a+1)^n+c)>1 && return(a))}

%o (Python)

%o from sympy import primefactors, resultant, nthroot_mod

%o from sympy.abc import m

%o def A255867(n):

%o     if n == 0: return 1

%o     k = 0

%o     for p in primefactors(resultant(m**n+17,(m+1)**n+17)):

%o         for d in (a for a in nthroot_mod(-17,n,p,all_roots=True) if pow(a+1,n,p)==-17%p):

%o             k = min(d,k) if k else d

%o     return k  # _Chai Wah Wu_, May 07 2024

%Y Cf. A118119, A255832, A255852-A255869

%K nonn,hard

%O 0,4

%A _M. F. Hasler_, Mar 09 2015

%E a(5)-a(22) from _Hiroaki Yamanouchi_, Mar 12 2015

%E a(23)-a(60) from _Max Alekseyev_, Aug 06 2015