A303745 Totients t where gcd({x: phi(x)=t}) > 1.

10, 22, 28, 30, 44, 46, 52, 54, 56, 58, 66, 70, 78, 82, 92, 102, 104, 106, 110, 116, 126, 130, 136, 138, 140, 148, 150, 164, 166, 172, 178, 184, 190, 196, 198, 204, 208, 210, 212, 220, 222, 226, 228, 238, 250, 260, 262, 268, 270, 282, 292, 294, 296, 306

Offset: 1

Author

Torlach Rush, Apr 29 2018

Keywords

• nonn

Comments

If the least solution of phi(x)=t is prime then gcd({x: phi(x)=t}) is prime.

If gcd({x: phi(x)=t}) > 1 is not prime then the least solution of phi(x)=t is not prime.

For known terms if the number of solutions of x: phi(x)=t is 2 or 3 then the least solution divides the greatest solution (see A297475). - Torlach Rush, Jul 03 2018

Examples

			10 is a term because the greatest common divisor of 11 and 22, the solutions of phi(10) is 11.
2 is not a term because the greatest common divisor of 3, 4 and 6, the solutions of phi(2) is 1.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• Maxim Rytin, Finding the Inverse of Euler Totient Function, Wolfram Library Archive, 1999.

Crossrefs

Cf. A000010, A002202, A032447, A297475, A007367.

Programs

Maple

filter:= proc(n) local L;
L:= numtheory:-invphi(n);
L <> [] and igcd(op(L)) > 1
end proc:
select(filter, [seq(i,i=2..1000, 2)]); # Robert Israel, Jun 26 2018

Mathematica

Select[Range[2, 1000, 2], GCD@@invphi[#] > 1&] (* Jean-François Alcover, Jan 31 2023, using Maxim Rytin's invphi program *)

PARI

isok(n) = gcd(invphi(n)) > 1; \\ Michel Marcus, May 13 2018

Formula

gcd({x: phi(x)=t}) > 1.