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A245498 Least base B >= 2 such that the repunit (B^n-1)/(B-1) of length n is not squarefree. 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A120647 A131635 A324554

Adjacent sequences:  A245495 A245496 A245497 * A245499 A245500 A245501

KEYWORD

nonn,more

AUTHOR

Jeppe Stig Nielsen, Jul 24 2014

STATUS

approved

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Last modified December 9 09:20 EST 2019. Contains 329877 sequences. (Running on oeis4.)