A307711 -id:A307711 - 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 *)

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

Original entry on oeis.org

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 *)

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica