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A172480
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Odd primes p such that there are as many primitive roots (mod p) in the interval [0,p/2] as in the interval [p/2,p].
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3
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5, 7, 13, 17, 29, 31, 37, 41, 43, 53, 61, 67, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 307, 313, 317, 337, 349, 353, 367, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 487, 509, 521, 541, 557, 569
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OFFSET
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1,1
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COMMENTS
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The sequence contains all the primes of the form 4m+1 (A002144).
The sequence also contains some primes of the form 4m+3 (see them in A172490).
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LINKS
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MAPLE
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filter:= proc(p) local m; uses NumberTheory;
if not isprime(p) then return false fi;
if p mod 4 = 1 then return true fi;
m:= Totient(Totient(p))/2;
PrimitiveRoot(p, ith=m+1)=PrimitiveRoot(p, greaterthan=floor(p/2))
end proc:
select(filter, [seq(i, i=5..1000, 2)]); # Robert Israel, Nov 23 2019
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MATHEMATICA
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<< NumberTheory`NumberTheoryFunctions` m = 2; s = {}; While[m < 10000, m++; p = Prime[m]; If[Mod[p, 4] == 1, s = {s, p}, q = (p - 1)/2; g = PrimitiveRoot[p]; se = Select[Range[p - 1], GCD[ #, p - 1] == 1 &]; e = Length[se]; j = 0; t = 0; While[j < e, j++; h = PowerMod[g, se[[j]], p]; If[h <= q, t = t + 1, ] ]; If[e == 2t, s = {s, p}, ] ] ]; s = Flatten[s]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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