A245498 Least base B >= 2 such that the repunit (B^n-1)/(B-1) of length n is not squarefree.

3, 18, 3, 3, 2, 78, 3, 4, 3, 118, 2, 146, 3, 3, 3, 164, 2, 44, 2, 2, 3, 53, 2, 3, 3, 4, 3, 53, 2, 403, 3, 18, 3, 3, 2, 957, 3, 3, 2, 99, 2, 369, 3, 3, 3, 533, 2, 8, 3, 18, 3, 164, 2, 3, 3, 7, 3, 381, 2

Offset

2,1

Comments

When n is prime, a(n) seems to be hard to determine.

Let p be a prime == 1 (mod n) (such a prime exists by Dirichlet's theorem). Since gcd(n, phi(p)) > 1 there exists b such that 1 < b < p and b^n == 1 (mod p). Then x = b + y*p for suitable y has x^n == 1 (mod p^2), and x == b (mod p), i.e., (x^n-1)/(x-1) is divisible by p^2. Therefore a(n) <= x < p^2. - Robert Israel, Jul 24 2014

Links

Table of n, a(n) for n=2..60.

Example

a(17)=164 because (164^17 - 1)/163 is not squarefree (is multiple of 103^2), and 164 is the minimal number with that property.

Maple

A:= proc(n) local x, F;
    for x from 2 do F:= ifactors((x^n-1)/(x-1), easy)[2];
      if max(seq(f[2], f=F)) >= 2
      then return x
    fi
   od
end proc;
seq(A(n), n=2..50); # Robert Israel, Jul 24 2014

Prog

(PARI) for(n=2, 100, b=2; while(issquarefree((b^n-1)/(b-1)), b++); print1(b, ", "))

Crossrefs

Sequence in context: A082056 A082057 A161687 * A350703 A120647 A131635

Adjacent sequences: A245495 A245496 A245497 * A245499 A245500 A245501

Keyword

nonn

Author

Jeppe Stig Nielsen, Jul 24 2014

Status

approved