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

%I A307712 #11 Apr 28 2019 23:03:26
%S A307712 3,4,5,6,7,9,10,15,21,31,45,49,58,65,82,86,92,97,101,105,116,183,187,
%T A307712 196,201,207,217,238,297,305,308,310,320,331,380,425,583,649,675,855,
%U A307712 964,972,974,978,993,996,998,1009,1016,1017,1041,1068,1093,1112,1117,1123,1129,1161,1184,1368,1403
%N A307712 Numbers k such that the fraction of primes in the reduced residue system mod k is the reciprocal of an integer.
%C A307712 Numbers k such that A000010(k)/A048865(k) is an integer.
%C A307712 The corresponding integers are in A307713.
%H A307712 Robert Israel, <a href="/A307712/b307712.txt">Table of n, a(n) for n = 1..600</a>
%e A307712 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.
%e A307712 8 is not in the sequence because 3 of the 4 reduced residues mod 8 are prime, and 3 does not divide 4.
%p A307712 filter:= proc(n) uses numtheory;
%p A307712 type(phi(n)/(pi(n) - nops(factorset(n))),integer);
%p A307712 end proc:
%p A307712 select(filter, [$3..10000]);
%t A307712 Select[Range[3, 1500], Function[n, IntegerQ[EulerPhi[n]/Count[Prime@ Range@ PrimePi@ n, _?(GCD[#, n] == 1 &)]]]] (* _Michael De Vlieger_, Apr 23 2019 *)
%Y A307712 Cf. A000010, A048865, A307711, A307713.
%K A307712 nonn
%O A307712 1,1
%A A307712 _Robert Israel_, Apr 23 2019