%I #17 Jul 28 2024 16:51:15
%S 1,3,6,7,6,3,0,8,0,1,9,8,5,0,2,2,3,5,0,7,9,0,5,0,8,1,4,6,2,1,3,0,8,8,
%T 1,3,9,0,7,4,8,9,1,9,9,8,9,6,2,7,9,4,8,5,2,9,5,6,5,9,8,4,6,3,7,6,2,1,
%U 5,6,7,1,0,3,9,7,6,6,8,7,4,4,5,5,0,3,7,9,0,0,7,0,5,4,2,8,2,8,0
%N Decimal expansion of Sum_{k>=1} phi(k)/2^k, where phi is Euler's totient function.
%D Richard K. Guy, Unsolved Problems in Number Theory, Springer, 2004, p. 139.
%H Paul Erdős and Ronald L. Graham, <a href="http://www.math.ucsd.edu/~fan/ron/papers/80_11_number_theory.pdf">Old and new problems and results in combinatorial number theory</a>, Monographies de L'Enseignement Mathematique, L'enseignement Mathématique, Université de Genève, 1980, p. 61.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>.
%F Equals Sum_{k>=1} A007431(k)/(2^k - 1). - _Amiram Eldar_, Jun 23 2020
%e 1.36763080198502235079050814621308813907489199896...
%t digits = 99; m0 = 10; dd = 10; Clear[f]; f[m_] := f[m] = Sum[EulerPhi[n]/2^n, {n, 1, m}] // N[#, digits + 2*dd]&; f[m = m0] ; While[RealDigits[f[2*m], 10, digits + dd ] != RealDigits[f[m], 10, digits + dd ], m = 2*m; Print[m]]; RealDigits[f[m], 10, digits] // First
%o (PARI) suminf(n=1,eulerphi(n)/2^n) \\ _Charles R Greathouse IV_, Apr 20 2016
%Y Cf. A000010, A007431.
%K nonn,cons
%O 1,2
%A _Jean-François Alcover_, Apr 13 2015