

A242248


a(n) = {0 < g < prime(n): g, 2^g  1 and (g1)! are all primitive roots modulo prime(n)}.


4



1, 0, 1, 1, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 6, 2, 10, 3, 2, 3, 5, 2, 10, 12, 3, 6, 7, 15, 3, 9, 3, 8, 18, 5, 18, 3, 7, 7, 24, 20, 26, 4, 13, 10, 15, 5, 4, 3, 35, 5, 19, 19, 3, 19, 36, 37, 38, 5, 10, 15, 16, 34, 7, 16, 6, 36, 4, 4, 44, 14
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OFFSET

1,7


COMMENTS

Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there exists a positive primitive root g < p modulo p such that 2^g  1 and (g1)! are also primitive roots modulo p.
We have verified this for all primes p with 3 < p < 10^6.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..1500
ZhiWei Sun, Notes on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.


EXAMPLE

a(4) = 1 since 5, 2^5  1 = 31 and (51)! = 24 are all primitive roots modulo prime(4) = 7.
a(6) = 1 since 11, 2^(11)  1 = 2047 and (111)! = 3628800 are all primitive roots modulo prime(6) = 13. Note that both 2047 and 3628800 are congruent to 6 modulo 13.
a(14) = 1 since 34, 2^(34)  1 and (341)! are all primititive roots modulo prime(14) = 43. Note that 2^(34)  1 == 20 (mod 43) and 33! == 14 (mod 43).


MATHEMATICA

f[n_]:=2^n1
g[n_]:=(n1)!
rMod[m_, n_]:=Mod[m, n, n/2]
dv[n_]:=Divisors[n]
Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa]]; Do[If[Mod[k^(Part[dv[Prime[n]1], i]), Prime[n]]==1Mod[rMod[f[k], Prime[n]]^(Part[dv[Prime[n]1], i]), Prime[n]]==1Mod[rMod[g[k], Prime[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, Prime[n]1}]; Print[n, " ", m]; Continue, {n, 1, 70}]


CROSSREFS

Cf. A000040, A000142, A000225, A236966, A237112, A242242, A242250.
Sequence in context: A010275 A176494 A157229 * A107297 A107296 A080847
Adjacent sequences: A242245 A242246 A242247 * A242249 A242250 A242251


KEYWORD

nonn


AUTHOR

ZhiWei Sun, May 09 2014


STATUS

approved



