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A143811
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Number of numbers k<p such that k^(p-1)-1 is divisible by p^2, p = prime(n).
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1
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1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 2, 1, 2, 2, 3, 3, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 1, 2, 2, 3, 2, 1, 3, 2, 1, 2, 1, 2, 2, 4, 1, 1, 2, 2, 2, 2, 1, 3, 1, 4, 1, 3, 3, 3, 3, 3, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 5, 1, 2, 1, 3, 2, 2, 1, 2, 2, 2, 1, 4
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OFFSET
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1,5
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COMMENTS
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Note that a(n)>0 because k=1 is always a solution. The primes for which a(n)>1 are given in A134307. The values of k are the terms <p in row n of A143548. The largest known terms in this sequence are for the Wieferich primes 1093 and 3511, for which we have a(183)=11 and a(490)=12, respectively. It is not hard to show that k=p-1 is never a solution for odd prime p. In fact, (p-1)^(p-1)=p+1 (mod p^2) for odd prime p.
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LINKS
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MATHEMATICA
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Table[p=Prime[n]; s=Select[Range[p-1], PowerMod[ #, p-1, p^2]==1&]; Length[s], {n, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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