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 A014197 Number of numbers m with Euler phi(m) = n. 30

%I

%S 2,3,0,4,0,4,0,5,0,2,0,6,0,0,0,6,0,4,0,5,0,2,0,10,0,0,0,2,0,2,0,7,0,0,

%T 0,8,0,0,0,9,0,4,0,3,0,2,0,11,0,0,0,2,0,2,0,3,0,2,0,9,0,0,0,8,0,2,0,0,

%U 0,2,0,17,0,0,0,0,0,2,0,10,0,2,0,6,0,0,0,6,0,0,0,3

%N Number of numbers m with Euler phi(m) = n.

%C Carmichael conjectured that there are no 1's in this sequence.

%C Number of cyclotomic polynomials of degree n. - _T. D. Noe_, Aug 15 2003

%D R. K. Guy, Unsolved Problems in Number Theory, section B39.

%D J. Roberts, Lure of The Integers, entry 32, page 182.

%H T. D. Noe, <a href="/A014197/b014197.txt">Table of n, a(n) for n = 1..10000</a>

%H K. Ford, <a href="http://arXiv.org/abs/math.NT/9907204">[math/9907204] The number of solutions of phi(x)=m</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientValenceFunction.html">Totient Valence Function</a>

%F Dirichlet g.f.: sum(n>=1, a(n)*n^-s)=zeta(s)*prod(1+1/(p-1)^s-1/p^s) - _Benoit Cloitre_, Apr 12 2003

%F lim n ->infinity (1/n)*sum(k=1, n, a(k))=zeta(2)*zeta(3)/zeta(6)=1.94359643682075920505707036... - _Benoit Cloitre_, Apr 12 2003

%p with(numtheory): A014197 := n-> nops(invphi(i));

%t inversePhi[m_?OddQ] = {}; inversePhi[1] = {1, 2}; inversePhi[m_] := Module[ {p, nmax, n, nn}, p = Select[ Divisors[m] + 1, PrimeQ]; nmax = m*Times @@ (p/(p - 1)); n = m; nn = {}; While[n <= nmax, If[ EulerPhi[n] == m, AppendTo[nn, n]]; n++]; nn]; a[n_] := Length[ inversePhi[n] ]; Table[ a[n], {n, 1, 92}] (* From Jean-François Alcover, Dec 09 2011 *)

%o (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 _M. F. Hasler_, Oct 05 2009]

%Y Cf. A058277, A002202, A032446.

%Y Cf. A070243 (partial sums).

%Y For records see A131934, A097942.

%K nonn,nice,easy

%O 1,1

%A _N. J. A. Sloane_.