A237594 Number of primes p < prime(n)/2 such that the Bell number B(p) is a primitive root modulo prime(n).

0, 0, 1, 1, 2, 1, 2, 3, 1, 1, 2, 4, 2, 4, 6, 5, 7, 3, 4, 3, 4, 2, 12, 7, 3, 5, 4, 9, 5, 6, 4, 5, 12, 6, 7, 5, 9, 6, 12, 11, 13, 7, 7, 7, 14, 5, 5, 14, 14, 8, 13, 11, 7, 10, 19, 17, 16, 8, 11, 7, 7, 23, 11, 12, 10, 22, 14, 8, 22, 11, 20, 22, 13, 13, 15, 24, 27, 14, 18, 18

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a prime q < p/2 such that the Bell number B(q) is a primitive root modulo p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014.

Example

a(9) = 1 since 3 is a prime smaller than prime(9)/2 = 23/2 and B(3) = 5 is a primitive root modulo prime(9) = 23.

Mathematica

f[k_]:=BellB[Prime[k]]
dv[n_]:=Divisors[n]
Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa], Do[If[Mod[f[k]^(Part[dv[Prime[n]-1], i]), Prime[n]]==1, Goto[aa]], {i, 1, Length[dv[Prime[n]-1]]-1}]]; m=m+1; Label[aa]; Continue, {k, 1, PrimePi[(Prime[n]-1)/2]}]; Print[n, " ", m]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000040, A000110, A236308, A236966, A237112, A237121.

Sequence in context: A348245 A215026 A162319 * A098666 A362890 A306251

Adjacent sequences: A237591 A237592 A237593 * A237595 A237596 A237597

Keyword

nonn

Author

Zhi-Wei Sun, Apr 22 2014

Status

approved