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A243164 Number of primes p < n such that p*n is a primitive root modulo prime(n). 6
0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 2, 3, 4, 2, 2, 4, 3, 3, 5, 3, 5, 5, 6, 2, 5, 5, 4, 4, 2, 4, 6, 7, 6, 7, 2, 7, 4, 7, 4, 4, 6, 7, 3, 7, 7, 3, 7, 7, 9, 7, 6, 5, 6, 6, 7, 7, 9, 4, 9, 8, 2, 10, 7, 9, 11, 5, 6, 5, 9, 11, 8, 6, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,11

COMMENTS

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

(ii) Any integer n > 6 can be written as k + m with k > 0 and m > 0 such that k*m is a primitive root modulo prime(n).

We have verified part (i) for all n = 7, ..., 2*10^5.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(4) = 1 since 3 is prime with 3*4 = 12 a primitive root modulo prime(4) = 7.

a(9) = 1 since 7 is prime with 7*9 = 63 a primitive root modulo prime(9) = 23.

a(10) = 1 since 5 is prime with 5*10 = 50 a primitive root modulo prime(10) = 29.

a(12) = 1 since 2 is prime with 2*12 = 24 a primitive root modulo prime(12) = 37.

MATHEMATICA

dv[n_]:=Divisors[n]

Do[m=0; Do[Do[If[Mod[(Prime[k]*n)^(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

Cf. A000040, A000720, A237497, A237578, A242748.

Sequence in context: A306433 A308174 A126237 * A333708 A132203 A158925

Adjacent sequences:  A243161 A243162 A243163 * A243165 A243166 A243167

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 31 2014

STATUS

approved

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Last modified August 3 04:47 EDT 2021. Contains 346435 sequences. (Running on oeis4.)