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A245498
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Least base B >= 2 such that the repunit (B^n-1)/(B-1) of length n is not squarefree.
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0
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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
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OFFSET
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2,1
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COMMENTS
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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
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LINKS
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Table of n, a(n) for n=2..60.
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EXAMPLE
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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.
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MAPLE
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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
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PROG
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(PARI) for(n=2, 100, b=2; while(issquarefree((b^n-1)/(b-1)), b++); print1(b, ", "))
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CROSSREFS
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Sequence in context: A082056 A082057 A161687 * A120647 A131635 A324554
Adjacent sequences: A245495 A245496 A245497 * A245499 A245500 A245501
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KEYWORD
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nonn,more
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AUTHOR
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Jeppe Stig Nielsen, Jul 24 2014
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STATUS
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approved
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