A179077 a(n) is the residue ((2^p - 2)/p) mod p, where p is the n-th prime.

1, 2, 1, 4, 10, 6, 9, 6, 11, 2, 12, 2, 5, 7, 41, 19, 16, 11, 20, 4, 39, 38, 13, 12, 17, 83, 15, 26, 25, 53, 36, 34, 106, 60, 43, 112, 7, 134, 94, 6, 100, 115, 100, 15, 153, 71, 7, 155, 175, 136, 14, 52, 43, 243, 193, 256, 251, 218, 140, 148, 116, 156, 281, 39, 240, 33, 278

Offset: 1

Author

Artur Jasinski, Jun 28 2010

Keywords

• nonn

Comments

a(n) = 0 where n=183 (p=1093) and n=490 (p=3511).

From Felix Fröhlich, Sep 13 2019: (Start)

Conjecture: a(n) is the residue A036968(p-1) (mod p) for p = prime(n).

If the above conjecture is true, then a(n) = 0 if and only if p is a Wieferich prime (A001220) (cf. Hu et al., 2019, section 1.3). (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Su Hu, Min-Soo Kim, Pieter Moree and Min Sha, Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture, arXiv:1809.08431 [math.NT], 2019; Journal of Number Theory 205 (2019), 59-80, DOI:10.1016/j.jnt.2019.03.012.

Crossrefs

Cf. A001220, A036968, A377655.

Programs

Maple

f:= p -> (2&^p-2 mod p^2)/p:
seq(f(ithprime(i)),i=1..100);# Robert Israel, Nov 03 2024

Mathematica

aa = {}; Do[AppendTo[aa, Mod[(2^Prime[n] - 2)/Prime[n], Prime[n]]], {n, 1, 100}]; aa

PARI

a(n) = my(p=prime(n)); lift(Mod(((2^p-2)/p), p)) \\ Felix Fröhlich, Sep 13 2019