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A303745 Totients t where gcd({x: phi(x)=t}) > 1. 6

%I #50 Jan 31 2023 11:48:12

%S 10,22,28,30,44,46,52,54,56,58,66,70,78,82,92,102,104,106,110,116,126,

%T 130,136,138,140,148,150,164,166,172,178,184,190,196,198,204,208,210,

%U 212,220,222,226,228,238,250,260,262,268,270,282,292,294,296,306

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

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

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

%C 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

%H Robert Israel, <a href="/A303745/b303745.txt">Table of n, a(n) for n = 1..10000</a>

%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI scripts for various problems</a>

%H Maxim Rytin, <a href="http://library.wolfram.com/infocenter/MathSource/696/">Finding the Inverse of Euler Totient Function</a>, Wolfram Library Archive, 1999.

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

%e 10 is a term because the greatest common divisor of 11 and 22, the solutions of phi(10) is 11.

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

%p filter:= proc(n) local L;

%p L:= numtheory:-invphi(n);

%p L <> [] and igcd(op(L)) > 1

%p end proc:

%p select(filter, [seq(i,i=2..1000, 2)]); # _Robert Israel_, Jun 26 2018

%t Select[Range[2, 1000, 2], GCD@@invphi[#] > 1&] (* _Jean-François Alcover_, Jan 31 2023, using Maxim Rytin's invphi program *)

%o (PARI) isok(n) = gcd(invphi(n)) > 1; \\ _Michel Marcus_, May 13 2018

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

%K nonn

%O 1,1

%A _Torlach Rush_, Apr 29 2018

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