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%I #28 Mar 15 2021 03:41:13
%S 2,4,7,6,1,9,4,7,4,8,1,6,5,0,2,5,7,9,4,3,2,6,8,5,5,4,4,4,1,2,5,1,4,5,
%T 1,6,0,0,4,5,4,5,6,8,5,6,3,5,5,2,8,4,3,8,4,3,4,5,7,0,7,8,7,9,1,5,0,9,
%U 4,9,0,3,0,1,1,7,5,1,2,4,5,8,1,7,6,2,8,0,1,3,4,6,1,5,2,6,7,3,8,9,3,3,2,8,9
%N Decimal expansion of Sum_{k>=1} 1/phi(k)^3, where phi is the Euler totient function.
%C Sum_{k>=1} 1/phi(k)^m is convergent iff m > 1 (reference Monier). - _Bernard Schott_, Jan 14 2021
%D Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294.
%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/TotientSummatoryFunction.html">Totient Summatory Function</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler%27s_phi_function">Euler's totient function</a>.
%F Equals Product_{primes p} (1 + 1/((1 - 1/p)^3 * (p^3 - 1))).
%e 2.476194748165025794326855444125145160045456856355284384345707879150949...
%t $MaxExtraPrecision = 1000; f[p_] := (1 + 1/((1 - 1/p)^s * (p^s - 1))) /. s -> 3; Do[cc = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[cc, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 120]]], {m, 100, 1000, 100}]
%o (PARI) prodeulerrat(1 + 1/((1 - 1/p)^3 * (p^3 - 1))) \\ _Amiram Eldar_, Mar 15 2021
%Y Cf. A000010, A065484, A109695, A335319.
%K nonn,cons
%O 1,1
%A _Vaclav Kotesovec_, Jun 25 2020
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