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A364883 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.

%I A364883 #30 Feb 16 2025 08:34:06
%S A364883 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 A364883 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 A364883 18,18,19,19,19,12,7,7,20,20,20,10,21,21,22,22,22,13,9,9,23,23,23
%N A364883 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 A364883 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 A364883 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 A364883 Chris Caldwell, PrimePages, <a href="https://t5k.org/glossary/page.php?sort=FermatQuotient">Fermat quotient</a>.
%H A364883 nLab, <a href="http://ncatlab.org/nlab/show/Fermat+quotient">Fermat quotient</a>.
%H A364883 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 A364883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatQuotient.html">Fermat Quotient</a>.
%H A364883 Wikipedia, <a href="http://en.wikipedia.org/Fermat_quotient">Fermat quotient</a>.
%e A364883 For a(2), examine A007663 and notice that beginning with the second term, offset is 2, all terms are divisible by 3;
%e A364883 For a(3), examine A146211 and notice that beginning with the first term, offset is 3, all terms are divisible by 8;
%e A364883 For a(4), examine A180511 and notice that beginning with the third term, offset is 2, all terms are divisible by 15; etc.
%t A364883 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 A364883 Cf. A007663, A096060, A146211, A180511.
%K A364883 nonn
%O A364883 2,1
%A A364883 _Robert G. Wilson v_, Aug 17 2023