A266986 The indices of primes p for which the average of the primitive roots equals p/2.

1, 3, 6, 7, 8, 10, 12, 13, 16, 18, 21, 24, 25, 26, 29, 30, 33, 35, 37, 40, 42, 44, 45, 50, 51, 53, 55, 57, 59, 60, 62, 63, 65, 66, 68, 70, 71, 74, 77, 78, 79, 80, 82, 84, 87, 88, 89, 97, 98, 100, 102, 104, 106, 108, 110, 112, 113, 116, 119, 121, 122, 123, 126, 127, 130, 134, 135, 136, 137, 139, 140, 142, 145

Offset

1,2

Comments

The average of the primitive roots of a prime p are <,=, or > p/2 (observation).

The indices of all primes p==1(mod 4) are in this sequence since for primes of form 4k+1 b a primitive root implies -b a primitive root.

The indices of some primes p==3 (mod 4) are also in this sequence although for most such primes the average of the primitive roots is <> p/2.(observation)

Links

Dimitri Papadopoulos, Table of n, a(n) for n = 1..505

Example

p(a(1))=p(1)=2. 2 has the primitive root 1. The average primitive root is 1 and 1=2/2.
p(a(2))=p(3)=5. The primitive roots of 5 are 2 and 3. Their average equals (2+3)/phi(4)=5/2=p/2.

Mathematica

A = Table[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[k]], {i, Prime[k] - 1}], Prime[k] - 1]]]/(EulerPhi[Prime[k] - 1] Prime[k]/2), {k, 1, 1000}]; Flatten[Position[A, _?(# == 1 &)]]

Crossrefs

Cf. A008330, A060749, A088144, A266987.

Sequence in context: A324143 A050244 A295566 * A047283 A155932 A374483

Adjacent sequences: A266983 A266984 A266985 * A266987 A266988 A266989

Keyword

nonn

Author

Dimitri Papadopoulos, Jan 08 2016

Status

approved