A242242 Least positive primitive root g < prime(n) modulo prime(n) with 2^g - 1 also a primitive root modulo prime(n), or 0 if such a number g does not exist.

1, 0, 2, 5, 2, 2, 3, 2, 5, 2, 3, 2, 6, 3, 5, 2, 2, 2, 2, 7, 5, 3, 2, 3, 5, 2, 5, 2, 6, 3, 3, 2, 3, 2, 2, 6, 5, 2, 5, 2, 2, 2, 19, 5, 2, 3, 2, 3, 2, 6, 3, 7, 7, 6, 3, 5, 2, 6, 5, 3, 3, 2, 5, 17, 10, 2, 3, 10, 2, 2

Offset

1,3

Comments

Conjecture: a(n) > 0 except for n = 2. In other words, for any prime p > 3, there exists a primitive root 0 < g < p modulo p such that 2^g - 1 is also a primitive root modulo p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Example

a(4) = 5 since both 5 and 2^5 - 1 = 31 are primitive roots modulo prime(4) = 7, but none of 1, 2, 4 and 2^3 - 1 is a primitive root modulo prime(4) = 7.

Mathematica

f[n_]:=f[n]=2^n-1
dv[n_]:=Divisors[n]
Do[Do[If[Mod[f[k], Prime[n]]==0, Goto[aa]]; Do[If[Mod[k^(Part[dv[Prime[n]-1], j])-1, Prime[n]]==0, Goto[aa]], {j, 1, Length[dv[Prime[n]-1]]-1}]; Do[If[rMod[f[k]^(Part[dv[Prime[n]-1], i])-1, Prime[n]]==0, Goto[aa]], {i, 1, Length[dv[Prime[n]-1]]-1}];
Print[n, " ", k]; Goto[bb]; Label[aa]; Continue, {k, 1, Prime[n]-1}]; Label[cc]; Print[n, " ", 0]; Label[bb]; Continue, {n, 1, 70}]

Crossrefs

Cf. A000040, A000225, A234972, A236966.

Sequence in context: A327867 A286664 A306541 * A171937 A065291 A065267

Adjacent sequences: A242239 A242240 A242241 * A242243 A242244 A242245

Keyword

nonn

Author

Zhi-Wei Sun, May 09 2014

Status

approved