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 A237112 Number of primes p < prime(n)/2 with p! a primitive root modulo prime(n). 7
 0, 0, 1, 0, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 4, 4, 6, 4, 2, 3, 3, 4, 7, 9, 5, 7, 5, 8, 4, 7, 6, 7, 8, 7, 11, 8, 9, 7, 13, 10, 16, 4, 7, 8, 13, 9, 8, 12, 17, 10, 14, 12, 4, 10, 14, 15, 18, 8, 9, 8, 8, 18, 6, 8, 7, 16, 9, 11, 21, 15 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n) > 0 for all n > 4. In other words, for any prime p > 7, there exists a prime q < p/2 such that q! is a primitive root modulo p. See also A236306 for a similar conjecture. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..600 Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014. EXAMPLE a(7) = 1 since 3 is a prime smaller than prime(7)/2 = 17/2 and 3! = 6 is a primitive root modulo prime(7) = 17. a(9) = 1 since  5 is a prime smaller than prime(9)/2 = 23/2 and 5! = 120 is a primitive root modulo prime(9) = 23. MATHEMATICA f[k_]:=(Prime[k])! dv[n_]:=Divisors[n] Do[m=0; Do[Do[If[Mod[f[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[(Prime[n]-1)/2]}]; Print[n, " ", m]; Continue, {n, 1, 70}] CROSSREFS Cf. A000040, A000142, A234972, A235709, A235712, A236306, A236308, A236966, A237121. Sequence in context: A006513 A105224 A261627 * A238013 A303940 A280134 Adjacent sequences:  A237109 A237110 A237111 * A237113 A237114 A237115 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 22 2014 STATUS approved

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Last modified July 25 10:07 EDT 2021. Contains 346289 sequences. (Running on oeis4.)