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.

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

Offset: 2

Author

Robert G. Wilson v, Aug 17 2023

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Jean Bourgain, Kevin Ford, Sergei V. Konyagin, and Igor E. Shparlinski, On the Divisibility of Fermat Quotients, Michigan Mathematical Journal, Vol. 59 (Aug 2010), pp. 313-328.
• Chris Caldwell, PrimePages, Fermat quotient.
• nLab, Fermat quotient.
• H. S. Vandiver, Fermat's Quotient and related arithmetic functions, Proceedings of the National Academy of Sciences of the United States of America, Vol. 31, 1945.
• Eric Weisstein's World of Mathematics, Fermat Quotient.
• Wikipedia, Fermat quotient.

Crossrefs

Cf. A007663, A096060, A146211, A180511.

Programs

Mathematica

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]