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A364883
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Consider the Fermat quotient for base n: Fq(n,k) = (n^(p - 1) - 1)/p, where p = prime(k), for k >= 1. a(n) is the least k >= 1 such that Fq(n,j) is divisible by n^2 - 1 for all j >= k.
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0
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3, 3, 4, 4, 5, 5, 5, 4, 6, 6, 7, 7, 7, 5, 8, 8, 9, 9, 9, 6, 10, 10, 10, 7, 7, 7, 11, 11, 12, 12, 12, 8, 8, 8, 13, 13, 13, 9, 14, 14, 15, 15, 15, 10, 16, 16, 16, 5, 8, 8, 17, 17, 17, 6, 9, 11, 18, 18, 19, 19, 19, 12, 7, 7, 20, 20, 20, 10, 21, 21, 22, 22, 22, 13, 9, 9, 23, 23, 23
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OFFSET
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2,1
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COMMENTS
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Conjecture: numbers appear in the sequence only a finite number of times. Terms appear in runs of length 1, 2, or 3, never more. The first time a term k appears is when the index is even. The terms appear for the first time in their natural order.
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LINKS
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EXAMPLE
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For a(2), examine A007663 and notice that beginning with the second term, offset is 2, all terms are divisible by 3;
For a(3), examine A146211 and notice that beginning with the first term, offset is 3, all terms are divisible by 8;
For a(4), examine A180511 and notice that beginning with the third term, offset is 2, all terms are divisible by 15; etc.
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MATHEMATICA
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a[n_] := Block[{k = Floor[(1/2.3) n^(87/100) + 100]}, While[p = Prime@ k; PowerMod[n, p - 1, (n^2 - 1)*p] == 1, k--]; ++k]; Array[a, 79, 2]
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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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