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A255867 Least m > 0 such that gcd(m^n+17, (m+1)^n+17) > 1, or 0 if there is no such m. 2
1, 0, 1, 1925, 1, 189812175, 1, 2, 1, 116, 1, 55508752881180794569675021, 1, 337276, 1, 230, 1, 162, 1, 2628, 1, 15, 1, 3604979675443168377172749, 1, 53, 1, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

LINKS

Table of n, a(n) for n=0..60.

FORMULA

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).

EXAMPLE

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.

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

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

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.

MAPLE

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

  q1:= m^n + 17;

  q2:= (m+1)^n + 17;

  r:= resultant(q1, q2, m);

  bestm:= infinity;

  for p in numtheory:-factorset(r) do

    A:= [msolve(q1, p)];

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

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

  od;

  if bestm = infinity then -1 else bestm fi

end proc:

f(0):= 1: f(1):=0:

map(f, [$1..26]); # Robert Israel, May 31 2019

MATHEMATICA

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 *)

PROG

(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))}

CROSSREFS

Cf. A118119, A255832, A255852-A255869

Sequence in context: A035768 A107564 A135648 * A326018 A202051 A283949

Adjacent sequences:  A255864 A255865 A255866 * A255868 A255869 A255870

KEYWORD

nonn,hard

AUTHOR

M. F. Hasler, Mar 09 2015

EXTENSIONS

a(5)-a(22) from Hiroaki Yamanouchi, Mar 12 2015

a(23)-a(60) from Max Alekseyev, Aug 06 2015

STATUS

approved

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Last modified October 19 03:10 EDT 2021. Contains 348073 sequences. (Running on oeis4.)