A242210 Number of primes p < prime(n) such that the Bernoulli number B_{p-1} is a primitive root modulo prime(n).

0, 0, 1, 2, 1, 2, 2, 1, 4, 2, 3, 6, 3, 2, 5, 6, 5, 7, 4, 6, 6, 10, 11, 12, 8, 10, 9, 12, 10, 13, 9, 9, 10, 10, 17, 11, 7, 11, 18, 22, 15, 11, 12, 15, 21, 15, 10, 15, 23, 18, 26, 15, 15, 22, 26, 22, 25, 19, 26, 22, 22, 20, 17, 23, 20, 28, 17, 18, 28, 22

Offset: 1

Author

Zhi-Wei Sun, May 07 2014

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a prime q < p such that the Bernoulli number B_{q-1} is a primitive root modulo p.

(ii) For any prime p > 13, there exists a prime q < p such that the Euler number E_{q-1} is a primitive root modulo p.

We have verified part (i) for n up to 4.2*10^5, and part (ii) for primes p below 10^6.

Examples

			a(4) = 2 since 3 is a primitive root modulo prime(4) = 7, and both B_{2-1} = - 1/2 and B_{5-1} = - 1/30 are congruent to 3 modulo 7.
a(5) = 1 since B_{3-1} = 1/6 == 2 (mod 11) with 2 a primitive root modulo prime(5) = 11.
a(8) = 1 since B_{17-1} = -3617/510 == -4 (mod 19) with -4 a primitive root modulo prime(8) = 19.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..500
• Zhi-Wei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A027641, A027642, A122045, A242193, A242194, A242213.

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[m=0;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}];m=m+1;Label[aa];Continue,{k,1,n-1}];Print[n," ",m];Continue,{n,1,70}]