A178815 - OEIS

A178815

First base of a nonzero Fermat quotient mod the n-th prime.

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

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OFFSET

1,1

COMMENTS

First number m coprime to p = p_n such that p does not divide q_p(m), where q_p(m) = (m^(p-1) - 1)/p is the Fermat quotient of p to the base m.

It is known that a(n) = O((log p_n)^2) as n -> oo. It is conjectured that a(n) = 3 if p_n is a Wieferich prime. See Section 1.1 in Ostafe-Shparlinski (2010).

Additional comments, references, links, and cross-refs are in A001220.

a(n) > 3 iff prime(n) is a term of both A001220 and A014127, i.e., iff A240987(n) = 2. - Felix Fröhlich, Jul 09 2016

LINKS

Table of n, a(n) for n=1..105.

A. Ostafe and I. Shparlinski, Pseudorandomness and Dynamics of Fermat Quotients, arXiv:1001.1504 [math.NT], 2010.

Wikipedia, Generalized Wieferich primes

FORMULA

a(n) = 2 if n > 1 and p_n is not a Wieferich prime A001220.

a(n) > 2 if p_n is a Wieferich prime.

A178844(n) = ((a(n)^(p-1) - 1)/p) mod p, where p = p_n.

EXAMPLE

p_1 = 2 and 2^2 divides 1^(2-1) - 1 = 0 but not 3^(2-1) - 1 = 2, so a(1) = 3.