A242213 Least prime p < prime(n) such that the Bernoulli number B_{p-1} is a primitive root modulo prime(n), or 0 if such a prime p does not exist.

0, 0, 2, 2, 3, 2, 3, 17, 2, 2, 7, 2, 3, 19, 2, 2, 3, 2, 17, 2, 7, 2, 3, 3, 13, 2, 2, 3, 3, 3, 3, 3, 3, 5, 2, 3, 3, 7, 2, 2, 3, 2, 2, 5, 2, 2, 5, 3, 3, 3, 3, 2, 7, 3, 3, 2, 2, 2, 3, 5, 11, 2, 13, 2, 11, 2, 5, 17, 3, 2

Offset: 1

Author

Zhi-Wei Sun, May 07 2014

Keywords

• nonn

Comments

According to the conjecture in A242210, a(n) should be positive for all n > 2.

Examples

			a(7) = 3 since 3 is a primitive root modulo prime(7) = 17 but 2 is not.
		

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.

Crossrefs

Cf. A000040, A027641, A027642, A242193, A242210.

Programs

Mathematica

rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
f[k_]:=BernoulliB[Prime[k]-1]
dv[n_]:=Divisors[n]
Do[Do[If[rMod[f[k],Prime[n]]==0,Goto[aa]];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," ",Prime[k]];Goto[bb];Label[aa];Continue,{k,1,n-1}];Print[n," ",0];Label[bb];Continue,{n,1,70}]