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A243403
Number of primes p < n such that p*(n-p) is a primitive root modulo prime(n).
5
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
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 6.
We have verified this for all n = 7, ..., 2*10^5.
LINKS
Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.
EXAMPLE
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.
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 04 2014
STATUS
approved