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%I #24 Sep 18 2023 09:01:19
%S 1,1,8,7,4,8,40,13,64,41,94,59,132,39,4,51,222,43,278,157,346,191,406,
%T 227,484,263,562,305,640,178,24,99,280,447,942,169,1052,278,1168,31,
%U 1282,689,1422,747,58,819,1686,99,1838,482
%N 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.
%C The corresponding denominators are given in A327341.
%C 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).
%C 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 .
%D Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963, p. 100, Satz 2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindFunction.html">Dedekind Function</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.
%F 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.
%F a(n) = numerator(A173290(n)/n^2). - _Amiram Eldar_, Nov 24 2022
%e 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, ...
%e The limit of r(n) for n-> infinity is A323669 = 0.759908877317533285829...
%e r(10^5) is approximatly 0.7599142240 (10 digits).
%t psi[0] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); a[n_] := Numerator[Sum[psi[k], {k, 1, n}]/n^2]; Array[a, 50] (* _Amiram Eldar_, Sep 03 2019 *)
%o (PARI) dpsi(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
%o a(n) = numerator(sum(k=1, n, dpsi(k))/n^2); \\ _Michel Marcus_, Sep 18 2023
%Y Cf. A001615, A173290, A323669, A327341 (denominators).
%K nonn,frac,easy
%O 1,3
%A _Wolfdieter Lang_, Sep 03 2019
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