A305058 Totients t such that the number of divisors of t equals the number of solutions of phi(x) = t.

6, 12, 80, 160, 312, 352, 928, 1760, 1792, 3264, 3960, 7104, 7648, 13680, 15984, 16224, 17760, 19712, 20352, 20800, 21088, 22368, 23184, 25728, 25888, 26240, 27072, 29664, 47952, 57312, 60048, 62976, 67072, 73152, 74368, 77664, 78144, 81568, 85056, 85392, 86688

Offset: 1

Author

Torlach Rush, May 24 2018

Keywords

• nonn

Comments

For known terms gcd({phi(x) = t}) = 1.

This is not always the case, the smallest counterexample being t=4598784, a term of A303745, which has gcd({phi(x) = t}) = 1997. - Daniel Suteu, Dec 01 2018

Conjecture: Every term divides one or more subsequent terms.

Numbers n for which A000005(n) = A014197(n), positions of zeros in A322019. - Antti Karttunen, Dec 01 2018

Examples

			6 is a term because the divisors of 6 are {1,2,3,6} and the solutions of phi(x) = 6 are {7,9,14,18}.
12 is a term because the divisors of 12 are {1,2,3,4,6,12} and the solutions of phi(x) = 12 are {13,21,26,28,36,42}.
		

Links

• Daniel Suteu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A014197, A320000, A322019, A303745.

Programs

Mathematica

A014197[1] = 2; A014197[m_?OddQ] = 0; A014197[m_] := Module[{p, nmax, n, k}, p = Select[ Divisors[m]+1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; k = 0; While[n <= nmax, If[EulerPhi[n] == m, k++]; n++]; k]; aQ[n_] := (DivisorSigma[0 ,n] == A014197[n]); Select[Range[1000], aQ] (* Amiram Eldar, Dec 02 2018 after Jean-François Alcover at A014197 *)

PARI

A014197(n, m=1) = { n==1 && return(1+(m<2)); my(p, q); sumdiv(n, d, if( d>=m && isprime(d+1), sum( i=0, valuation(q=n\d, p=d+1), A014197(q\p^i, p))))}; \\ From A014197
isA305058(n) = (numdiv(n) == A014197(n)); \\ Antti Karttunen, Dec 01 2018

Perl

use ntheory ':all'; for (1..10**5) { print "$\n" if inverse_totient($) == divisor_sum($, 0) } # _Daniel Suteu, Dec 01 2018

Formula

tau(a(n)) = #{phi(x) = a(n)}.

Extensions

More terms from Daniel Suteu, Dec 01 2018