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 A109695 Decimal expansion of Sum_{n>=1} 1/phi(n)^2. 5
 3, 3, 9, 0, 6, 4, 2, 0, 0, 5, 5, 7, 2, 5, 0, 3, 9, 1, 6, 1, 4, 2, 5, 9, 5, 6, 6, 3, 0, 0, 2, 6, 3, 0, 7, 9, 3, 7, 4, 0, 5, 3, 7, 3, 8, 1, 2, 1, 4, 4, 7, 1, 6, 9, 1, 1, 8, 0, 7, 3, 9, 8, 1, 5, 6, 8, 5, 7, 3, 8, 1, 3, 1, 1, 1, 7, 7, 6, 3, 3, 2, 1, 3, 6, 5, 0, 4, 1, 0, 2, 4, 4, 4, 9, 5, 2, 3, 7, 4, 2, 9, 8, 2, 5, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The logarithm of the value can be expanded in a series Sum_{j>=2} c(j)*P(j) = P(2) + 2*P(3) + (7/2)*P(4) + ... where P(.) is the prime zeta function. The partial sums of the series are a slowly oscillating function of the upper limit of j, from which the bracketing interval [3.390642005572503655..., 3.390642005572504756...] for the constant can be computed. - R. J. Mathar, Feb 03 2009 Sum_{n>=1} 1/phi(n)^k is convergent iff k > 1 (reference Monier). - Bernard Schott, Dec 13 2020 REFERENCES Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.2.21, pp. 281 and 294. LINKS Table of n, a(n) for n=1..105. FORMULA Equals Product_p Sum_{k>=0} 1/phi(p^k)^2 = Product_p (1 + p^2/((p-1)^2*(p^2-1))). Equals Sum{n>=1} 1/A127473(n). - Amiram Eldar, Mar 15 2021 EXAMPLE 3.39064200557250391614259566300263079374053738121447169118... MATHEMATICA \$MaxExtraPrecision = 1000; f[p_] := (1 + p^2/((p - 1)^2*(p^2 - 1))); 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}] (* Vaclav Kotesovec, Jun 25 2020 *) PROG (PARI) my(N=1000000000); prodeuler(p=2, N, 1.+p^2/((p-1)^2*(p^2-1)))*(1+1/(N*log(N))) (PARI) prodeulerrat(1 + p^2/((p-1)^2*(p^2-1))) \\ Amiram Eldar, Mar 15 2021 CROSSREFS Cf. A000010, A065484, A127473, A335818. Sequence in context: A221225 A119006 A248241 * A372038 A010610 A140059 Adjacent sequences: A109692 A109693 A109694 * A109696 A109697 A109698 KEYWORD nonn,cons AUTHOR Franklin T. Adams-Watters, Aug 07 2005 EXTENSIONS Four more digits from R. J. Mathar, Feb 03 2009, 25 more Dec 18 2010 More digits from Vaclav Kotesovec, Jun 25 2020 STATUS approved

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Last modified April 18 22:18 EDT 2024. Contains 371782 sequences. (Running on oeis4.)