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A267010 Primes of the form p==3 (mod 4) such that the average of their primitive roots equals p/2. 2
19, 307, 1451, 2179, 2251, 2683, 2843, 3259, 3907, 4447, 11863, 12907, 17623, 30763, 37963, 51059, 52543, 86131, 92467, 104851, 129763, 131203, 146683, 150151, 156151, 156703, 162523, 163819, 174007, 245899, 263827, 287731, 348643, 353611, 400123, 412831, 423091, 432587 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Most primes for which the average of the primitive roots=p/2 are of the form p==1(mod 4). Much rarer for primes of form p==3(mod 4) to have this property. (Observation)

LINKS

Table of n, a(n) for n=1..38.

EXAMPLE

a(1)=19. The primitive roots of 19 are 2, 3, 10, 13, 14, and 15. Their average is (2+3+10+13+14+15)/phi(18)=57/phi(18)=57/6=19/2.

MATHEMATICA

f[n_] := If[Total[Flatten[Position[Table[MultiplicativeOrder[i, Prime[n]], {i, Prime[n] - 1}],    Prime[n] - 1]]] == EulerPhi[Prime[n] - 1]*Prime[n]/2, 1, 0];

For[k = 1, k < 10000, k++, If[f[k] == 1 && Mod[Prime[k], 4] == 3, Print[k, "  ", Prime[k]]]]

PROG

(PARI) vr(p) = j=0; r=vector(eulerphi(p-1)); pr=znprimroot(p); for(i=1, p-1, if(gcd(i, p-1)==1, r[j++]=lift(pr^i))); r; \\ after A060749

isok(p) = ((p % 4 == 3) && (vpr = vr(p)) && (vecsum(vpr) == #vpr*p/2)); \\ Michel Marcus, Jan 09 2016

CROSSREFS

Cf. A060749. Intersection of A002145 and A266987.

Sequence in context: A051562 A324359 A074460 * A182460 A032631 A142430

Adjacent sequences:  A267007 A267008 A267009 * A267011 A267012 A267013

KEYWORD

nonn

AUTHOR

Dimitri Papadopoulos, Jan 08 2016

EXTENSIONS

a(16)-a(38) from Michel Marcus, Jan 09 2016

STATUS

approved

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Last modified March 24 06:56 EDT 2019. Contains 321444 sequences. (Running on oeis4.)