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 A236966 Number of primes p < prime(n)/2 such that 2^p - 1 is a primitive root modulo prime(n). 13
 0, 0, 1, 1, 1, 1, 3, 2, 1, 3, 2, 1, 2, 2, 5, 6, 3, 4, 3, 5, 4, 5, 7, 9, 3, 5, 2, 10, 7, 7, 7, 7, 9, 5, 10, 4, 5, 7, 12, 11, 14, 6, 7, 5, 10, 9, 8, 5, 12, 15, 14, 8, 12, 11, 16, 12, 16, 9, 12, 10, 10, 14, 15, 10, 12, 14, 9, 10, 21, 9, 22, 21, 11, 9, 18, 24, 20, 17, 17, 16 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Conjecture: a(n) > 0 for all n > 2. In other words, for any prime p > 3, there is a prime q < p/2 with the Mersenne number 2^q - 1 a primitive root modulo p. We have verified this for all n = 3, ..., 530000. See also the comment in A234972. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..1200 Z.-W. Sun, New observations on primitive roots modulo primes, arXiv preprint arXiv:1405.0290 [math.NT], 2014. EXAMPLE a(12) = 1 since 17 is a prime smaller than prime(12)/2 = 37/2 with 2^(17) - 1 = 131071 a primitive root modulo prime(12) = 37. MATHEMATICA f[k_]:=2^(Prime[k])-1 dv[n_]:=Divisors[n] Do[m=0; Do[If[Mod[f[k], Prime[n]]==0, Goto[aa], 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, 80}] CROSSREFS Cf. A000040, A001348, A001918, A234972, A235709, A235712, A236306, A236308. Sequence in context: A130827 A070309 A287556 * A280048 A119910 A130784 Adjacent sequences:  A236963 A236964 A236965 * A236967 A236968 A236969 KEYWORD nonn AUTHOR Zhi-Wei Sun, Apr 22 2014 STATUS approved

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Last modified July 25 12:15 EDT 2021. Contains 346290 sequences. (Running on oeis4.)