A307711 a(n) is the least number k such that exactly fraction 1/n of the members of the reduced residue system mod k are prime, or 0 if there is no such k.

3, 31, 97, 331, 1009, 3067, 11513, 27403, 64621, 185617, 480853, 1333951, 3524431, 9558361, 26080333, 70411483, 189961939

Offset: 2

Author

J. M. Bergot and Robert Israel, Apr 23 2019

Keywords

• nonn
• more

Comments

a(n) is the least number k, if any exists, such that A000010(k)/A048865(k) = n.

a(n) = A307712(m) for the least m such that A307713(m)=n.

Examples

			Of the 30 members of the reduced residue system mod 31, exactly one-third, namely 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, are prime.  31 is the least number with this property, so a(3) = 31.
		

Crossrefs

Cf. A000010, A048865, A307712.

Programs

Maple

f:= proc(n) uses numtheory;
  phi(n)/(pi(n) - nops(factorset(n)));
end proc:
N:= 13: # to get a(2)..a(N)
R:= Array(2..N): count:= 0:
for k from 3 while count < N-1 do
  v:= f(k);
  if v::integer and v <= N and R[v] = 0 then
     R[v]:= k;
     count:= count+1;
  fi
od:
convert(R,list);

Mathematica

With[{s = Table[EulerPhi[n]/Count[Prime@ Range@ PrimePi@ n, ?(GCD[#, n] == 1 &)], {n, 3, 10^4}]}, Array[2 + FirstPosition[s, #][[1]] &, Max@ Select[s, IntegerQ] - 1, 2]] (* _Michael De Vlieger, Apr 23 2019 *)

Formula

n*A048865(a(n)) = A000010(a(n)).