A377655 -id:A377655 - cp's OEIS frontend

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

Artur Jasinski, Jun 28 2010

nonn

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)

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.

Cf. A001220, A036968, A377655.

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

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

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

A377669 a(n) is the least prime p such that (3^p - 3)/p == n (mod p), or -1 if there is no such prime p.

Original entry on oeis.org

11, 2, 3, 5, 7, 23, 43, 5721619, 2311, 105830189, 31300663, 13, 113, 17, 821, 1181, 19, 37

Offset: 0

Robert Israel, Nov 03 2024

For n = 18, 24, 27, 28, 30, 38, ..., a(n) > 6 * 10^9 if it is not -1.

a(18) > 2*10^11 if it is not -1. - Michael S. Branicky, Nov 04 2024

			a(4) = 7 because (3^7 - 3)/7 = 312 == 4 (mod 7), and 7 is the first prime that works.

Cf. A179078, A377655.

f:= p -> (3&^p-3 mod p^2)/p:
V:= Array(0..17): count:= 0:
p:= 1:
for i from 1 while count < 23 do
  p:= nextprime(p);
  v:= f(p);
  if v <= 22 and V[v] = 0 then V[v]:= i; count:= count+1 fi;
od:
convert(V, list);

lpp[n_]:=Module[{p=2},While[Mod[(3^p-3)/p,p]!=n,p=NextPrime[p]];p]; Array[lpp,17,0] (* Harvey P. Dale, Jun 07 2025 *)

a(n) = prime(i) where A179078(i) = n, if such i exists.

cp's OEIS Frontend

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

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