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

2,1

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.

Links

Table of n, a(n) for n=2..18.

Formula

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

Example

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.

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 *)

Crossrefs

Cf. A000010, A048865, A307712.

Sequence in context: A152417 A182232 A294973 * A077547 A104312 A326736

Adjacent sequences: A307708 A307709 A307710 * A307712 A307713 A307714

Keyword

nonn,more

Author

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

Status

approved