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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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%I #29 Oct 13 2023 23:29:14

%S 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,

%T 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,

%U 18,18,19,19,19,12,7,7,20,20,20,10,21,21,22,22,22,13,9,9,23,23,23

%N 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.

%C 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.

%H Jean Bourgain, Kevin Ford, Sergei V. Konyagin, and Igor E. Shparlinski, <a href="https://researchers.mq.edu.au/en/publications/on-the-divisibility-of-fermat-quotients">On the Divisibility of Fermat Quotients</a>, Michigan Mathematical Journal, Vol. 59 (Aug 2010), pp. 313-328.

%H Chris Caldwell, PrimePages, <a href="https://t5k.org/glossary/page.php?sort=FermatQuotient">Fermat quotient</a>.

%H nLab, <a href="http://ncatlab.org/nlab/show/Fermat+quotient">Fermat quotient</a>.

%H H. S. Vandiver, <a href="https://doi.org/10.1073/pnas.31.1.55">Fermat's Quotient and related arithmetic functions</a>, Proceedings of the National Academy of Sciences of the United States of America, Vol. 31, 1945.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FermatQuotient.html">Fermat Quotient</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/Fermat_quotient">Fermat quotient</a>.

%e For a(2), examine A007663 and notice that beginning with the second term, offset is 2, all terms are divisible by 3;

%e For a(3), examine A146211 and notice that beginning with the first term, offset is 3, all terms are divisible by 8;

%e For a(4), examine A180511 and notice that beginning with the third term, offset is 2, all terms are divisible by 15; etc.

%t 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]

%Y Cf. A007663, A096060, A146211, A180511.

%K nonn

%O 2,1

%A _Robert G. Wilson v_, Aug 17 2023