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%I #32 Oct 11 2022 05:49:52
%S 6,12,80,160,312,352,928,1760,1792,3264,3960,7104,7648,13680,15984,
%T 16224,17760,19712,20352,20800,21088,22368,23184,25728,25888,26240,
%U 27072,29664,47952,57312,60048,62976,67072,73152,74368,77664,78144,81568,85056,85392,86688
%N Totients t such that the number of divisors of t equals the number of solutions of phi(x) = t.
%C For known terms gcd({phi(x) = t}) = 1.
%C 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
%C Conjecture: Every term divides one or more subsequent terms.
%C Numbers n for which A000005(n) = A014197(n), positions of zeros in A322019. - _Antti Karttunen_, Dec 01 2018
%H Daniel Suteu, <a href="/A305058/b305058.txt">Table of n, a(n) for n = 1..10000</a>
%F tau(a(n)) = #{phi(x) = a(n)}.
%e 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}.
%e 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}.
%t 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 *)
%o (Perl) use ntheory ':all'; for (1..10**5) { print "$_\n" if inverse_totient($_) == divisor_sum($_, 0) } # _Daniel Suteu_, Dec 01 2018
%o (PARI)
%o 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
%o isA305058(n) = (numdiv(n) == A014197(n)); \\ _Antti Karttunen_, Dec 01 2018
%Y Cf. A000005, A000010, A014197, A320000, A322019, A303745.
%K nonn
%O 1,1
%A _Torlach Rush_, May 24 2018
%E More terms from _Daniel Suteu_, Dec 01 2018
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