A178844 First nonzero Fermat quotient mod the n-th prime.

1, 1, 3, 2, 5, 3, 13, 3, 17, 1, 6, 1, 23, 25, 44, 36, 8, 36, 10, 2, 56, 19, 48, 6, 57, 92, 59, 13, 67, 83, 18, 17, 53, 30, 96, 56, 82, 67, 47, 3, 50, 148, 50, 104, 175, 135, 109, 189, 201, 68, 7, 26, 142, 247, 225, 128, 260, 109, 70, 74, 58, 78, 294, 175, 120, 175, 139, 153

Offset: 1

Author

Jonathan Sondow, Jun 24 2010

Keywords

• nonn

Comments

First nonzero value of q_p(m) mod p with gcd(m,p) = 1, where q_p(m) = (m^(p-1) - 1)/p is the Fermat quotient of p to the base m and p is the n-th prime p_n.

It is believed that a(n) = q_p(3) mod p, if p = p_n is a Wieferich prime A001220. See Section 1.1 in Ostafe-Shparlinski (2010).

See additional comments, references, links, and cross-refs in A001220 and A178815.

Examples

			p_1 = 2 and (m^1 - 1)/2 = 0, 1 == 0, 1 (mod 2) for m = 1, 3, so a(1) = 1.
p_5 = 11 and (m^10 - 1)/11 = 0, 93 == 0, 5 (mod 7) for m = 1, 2, so a(4) = 5.
p_183 = 1093 and (m^1092 - 1)/1093 == 0, 0, 312 (mod 1093) for m = 1, 2, 3, so a(183) = 312.
Similarly, a(490) = 7.
		

Links

• A. Ostafe and I. Shparlinski (2010), Pseudorandomness and Dynamics of Fermat Quotients

Crossrefs

Cf. A001220, A130912, A178815.

Formula

a(n) = q_p(A178815(n)) mod p, where p = p_n.

a(n) = A130912(n), if n > 1 and p_n is not a Wieferich prime. (Note: the offset of A130912 is n = 2.)

Extensions

Nonexistent A-numbers removed by Jonathan Sondow, Jun 26 2010