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 A327340 Numerator of the rationals r(n) = (1/n^2)*Phi_1(n), with Phi_1(n) = Sum{k=1..n} psi(k), with Dedekind's psi function. 2
 1, 1, 8, 7, 4, 8, 40, 13, 64, 41, 94, 59, 132, 39, 4, 51, 222, 43, 278, 157, 346, 191, 406, 227, 484, 263, 562, 305, 640, 178, 24, 99, 280, 447, 942, 169, 1052, 278, 1168, 31, 1282, 689, 1422, 747, 58, 819, 1686, 99, 1838, 482 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The corresponding denominators are given in A327341. Dedekind's psi(k) = k*Product_{p|k}(1 + 1/p), with primes p, and the empty product is set to 1. See psi(k) = A001615(k), k >= 1. In the Walfisz reference psi(k) = phi_1(k). In the  Walfisz reference, Satz 2., p. 100, the approximation for Phi_1(x) = (15/(2*Pi^2))*x^2 + O(x*(log(x))^{2/3}) is given (with B instead of the O() notation). For the constant 15/(2*Pi^2) see A323669 . REFERENCES Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 100, Satz 2. LINKS Eric Weisstein's World of Mathematics, Dedekind Function Wikipedia, Dedekind psi function FORMULA a(n) = numerator(r(n)), with the rationals r(n) = (1/n^2)*Sum{k=1..n}(k*Product_{p|k}(1 + 1/p)), with distinct prime p divisors of k (with empty product set to 1 for k = 1), for n >= 1. EXAMPLE The rationals (in lowest terms) begin: 1/1, 1/1, 8/9, 7/8, 4/5, 8/9, 40/49, 13/16, 64/81, 41/50, 94/121, 59/72, 132/169, 39/49, 4/5, 51/64, 222/289, 43/54, 278/361, 157/200, 346/441, 191/242, 406/529, 227/288, 484/625, 263/338, 562/729, 305/392, 640/841, 178/225, 24/31, ... The limit of r(n) for n-> infinity is A327341  = 0.759908877317533285829... r(10^5) is approximatly 0.7599142240 (10 digits). MATHEMATICA psi = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][]); a[n_] := Numerator[Sum[psi[k], {k, 1, n}]/n^2]; Array[a, 50] (* Amiram Eldar, Sep 03 2019 *) CROSSREFS Cf. A001615, A323669, A327341 (denominators). Sequence in context: A070702 A019870 A019903 * A167222 A076417 A114137 Adjacent sequences:  A327337 A327338 A327339 * A327341 A327342 A327343 KEYWORD nonn,frac,easy AUTHOR Wolfdieter Lang, Sep 03 2019 STATUS approved

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Last modified September 24 06:13 EDT 2021. Contains 347623 sequences. (Running on oeis4.)