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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.
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%I #17 Apr 28 2019 20:09:04

%S 3,31,97,331,1009,3067,11513,27403,64621,185617,480853,1333951,

%T 3524431,9558361,26080333,70411483,189961939

%N 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.

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

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

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

%e 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.

%p f:= proc(n) uses numtheory;

%p phi(n)/(pi(n) - nops(factorset(n)));

%p end proc:

%p N:= 13: # to get a(2)..a(N)

%p R:= Array(2..N): count:= 0:

%p for k from 3 while count < N-1 do

%p v:= f(k);

%p if v::integer and v <= N and R[v] = 0 then

%p R[v]:= k;

%p count:= count+1;

%p fi

%p od:

%p convert(R,list);

%t 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 *)

%Y Cf. A000010, A048865, A307712.

%K nonn,more

%O 2,1

%A _J. M. Bergot_ and _Robert Israel_, Apr 23 2019