A243403 Number of primes p < n such that p*(n-p) is a primitive root modulo prime(n).

0, 0, 1, 1, 2, 0, 3, 2, 3, 2, 1, 3, 3, 2, 3, 4, 4, 1, 4, 1, 2, 2, 5, 8, 5, 1, 1, 5, 3, 6, 6, 7, 6, 6, 4, 2, 4, 3, 6, 11, 6, 4, 3, 7, 6, 8, 3, 2, 10, 9, 6, 11, 2, 8, 9, 9, 5, 2, 5, 2, 3, 13, 5, 14, 8, 12, 7, 8, 9, 6, 13, 9, 4, 10, 3, 13, 12, 4, 8, 4

Offset: 1

Author

Zhi-Wei Sun, Jun 04 2014

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 6.

We have verified this for all n = 7, ..., 2*10^5.

Examples

			a(18) = 1 since 17 is prime with 17*(18-17) = 17 a primitive root modulo prime(18) = 61.
a(20) = 1 since 11 is prime with 11*(20-11) = 99 a primitive root modulo prime(20) = 71.
a(26) = 1 since 2 is prime with 2*(26-2) = 48 a primitive root modulo prime(26) = 101.
a(27) = 1 since 17 is prime with 17*(27-17) = 170 a primitive root modulo prime(27) = 103.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Crossrefs

Cf. A000040, A000720, A243164.

Programs

Mathematica

dv[n_]:=Divisors[n]
Do[m=0;Do[Do[If[Mod[(Prime[k]*(n-Prime[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[n-1]}];
Print[n," ",m];Continue,{n,1,80}]