A243403 - OEIS

%I #14 Aug 05 2019 03:03:35

%S 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,

%T 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,

%U 12,7,8,9,6,13,9,4,10,3,13,12,4,8,4

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

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

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

%H Zhi-Wei Sun, <a href="/A243403/b243403.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1405.0290">New observations on primitive roots modulo primes</a>, arXiv:1405.0290 [math.NT], 2014.

%e a(18) = 1 since 17 is prime with 17*(18-17) = 17 a primitive root modulo prime(18) = 61.

%e a(20) = 1 since 11 is prime with 11*(20-11) = 99 a primitive root modulo prime(20) = 71.

%e a(26) = 1 since 2 is prime with 2*(26-2) = 48 a primitive root modulo prime(26) = 101.

%e a(27) = 1 since 17 is prime with 17*(27-17) = 170 a primitive root modulo prime(27) = 103.

%t dv[n_]:=Divisors[n]

%t 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]}];

%t Print[n," ",m];Continue,{n,1,80}]

%Y Cf. A000040, A000720, A243164.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Jun 04 2014