A307713 -id:A307713 - cp's OEIS frontend

A307712 Numbers k such that the fraction of primes in the reduced residue system mod k is the reciprocal of an integer.

3, 4, 5, 6, 7, 9, 10, 15, 21, 31, 45, 49, 58, 65, 82, 86, 92, 97, 101, 105, 116, 183, 187, 196, 201, 207, 217, 238, 297, 305, 308, 310, 320, 331, 380, 425, 583, 649, 675, 855, 964, 972, 974, 978, 993, 996, 998, 1009, 1016, 1017, 1041, 1068, 1093, 1112, 1117, 1123, 1129, 1161, 1184, 1368, 1403

Offset: 1

Robert Israel, Apr 23 2019

nonn

Numbers k such that A000010(k)/A048865(k) is an integer.

The corresponding integers are in A307713.

			a(6)=9 is in the sequence because 3 of the 6 reduced residues mod 9 are prime, and 3 divides 6. The reduced residues are 1,2,4,5,7,8, of which 2,5,7 are prime.
8 is not in the sequence because 3 of the 4 reduced residues mod 8 are prime, and 3 does not divide 4.

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

Cf. A000010, A048865, A307711, A307713.

filter:= proc(n) uses numtheory;
type(phi(n)/(pi(n) - nops(factorset(n))),integer);
end proc:
select(filter, [$3..10000]);

Select[Range[3, 1500], Function[n, IntegerQ[EulerPhi[n]/Count[Prime@ Range@ PrimePi@ n, ?(GCD[#, n] == 1 &)]]]] (* _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

A307712 Numbers k such that the fraction of primes in the reduced residue system mod k is the reciprocal of an integer.

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