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

1,4

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A029290 A115311 A035436 * A035369 A129719 A062602

Adjacent sequences: A242207 A242208 A242209 * A242211 A242212 A242213

Keyword

nonn

Author

Zhi-Wei Sun, May 07 2014

Status

approved