A237121 Number of primes p < prime(n)/2 such that P(p) is a primitive root modulo prime(n), where P(.) is the partition function given by A000041.

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

Offset

1,5

Comments

Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there is a prime q < p/2 with P(q) = A000041(q) a primitive root modulo p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Example

a(14) = 1 since 3 is a prime smaller than prime(14)/2 = 43/2 and P(3) = 3 is a primitive root modulo prime(14) = 43.

Mathematica

f[k_]:=PartitionsP[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, 70}]

Crossrefs

Cf. A000040, A000041, A236308, A236966, A237112.

Sequence in context: A369764 A369987 A293440 * A329493 A139821 A248972

Adjacent sequences: A237118 A237119 A237120 * A237122 A237123 A237124

Keyword

nonn

Author

Zhi-Wei Sun, Apr 22 2014

Status

approved