A307712 -id:A307712 - cp's OEIS frontend

A307713 a(n) = A000010(A307712(n))/A048865(A307712(n)).

2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 4, 4, 2, 2, 3, 4, 2, 3, 3, 4, 2, 3, 4, 2, 2, 2, 5, 2, 4, 5, 5, 3, 3, 3, 2, 3, 2, 4, 2, 3, 6, 3, 4, 4, 2, 6, 3, 6, 6, 6, 4, 3, 2, 6, 6, 3, 3, 6, 6, 7, 3, 3, 6, 4, 6, 3, 6, 7, 2, 6, 4, 3, 6, 4, 4, 4, 4, 5, 3, 7, 4, 2, 8, 4, 4, 4, 4, 4, 3, 4, 5, 6

Offset: 1

Robert Israel, Apr 23 2019

nonn

1/a(n) is the fraction of primes in the reduced residue system mod A307712(n).

			a(3) = 2 because A307712(3) = 5 and 1/2 of the reduced residue system mod 5 are primes.

Robert Israel, Table of n, a(n) for n = 1..600

Cf. A000010, A048865, A307711, A307712.

g:=  proc(n) uses numtheory;
local v;
v:= phi(n)/(pi(n) - nops(factorset(n)));
if v::integer then v fi;
end proc:
map(g, [$3..40000]);

Select[Table[EulerPhi[n]/Count[Prime@ Range@ PrimePi@ n, ?(GCD[#, n] == 1 &)], {n, 3, 25000}], IntegerQ] (* _Michael De Vlieger, Apr 23 2019 *)

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.

Original entry on oeis.org

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

Offset: 2

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

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.

			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.

Cf. A000010, A048865, A307712.

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);

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 *)

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

cp's OEIS Frontend

A307713 a(n) = A000010(A307712(n))/A048865(A307712(n)).

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