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%I #6 Feb 19 2021 02:00:12
%S 1,3,8,6,9,2,4,1,7,0,4,1,3,5,6,5,8,6,8,9,8,8,1,4,9,1,9,7,6,6,5,1,0,6,
%T 8,3,6,1,6,5,2,6,2,0,7,8,2,6,3,9,2,9,9,1,7,4,1,1,3,7,0,1,5,8,1,3,7,2,
%U 6,0,2,1,5,6,6,1,7,6,7,9,2,2,6,3,4,1,2
%N Decimal expansion of zeta(3) * zeta(4) * Product_{p prime} (1 + 1/p^2 - 2/p^3 - 2/p^5 + 2/p^6).
%C The constant c in the asymptotic formulas Sum_{n1, n2 <= x} sigma(lcm(n1, n2)) = c * x^4/4 + O(x^(7/2 + eps)) and Sum_{n1, n2 <= x} sigma(lcm(n1, n2))/(n1*n2) = c * x^2 + O(x^(3/2 + eps)).
%H Titus Hilberdink and László Tóth, <a href="https://doi.org/10.1016/j.jnt.2016.05.024">On the average value of the least common multiple of k positive integers</a>, Journal of Number Theory, Vol. 169 (2016), pp. 327-341. See p. 333.
%e 1.38692417041356586898814919766510683616526207826392...
%t $MaxExtraPrecision = 1500; m = 1500; c = LinearRecurrence[{0, -1, 2, 0, 2, -2}, {0, 2, -6, -2, 0, 2}, m]; Zeta[3] * Zeta[4] * Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]]
%o (PARI) zeta(3) * zeta(4) * prodeulerrat(1 + 1/p^2 - 2/p^3 - 2/p^5 + 2/p^6)
%Y Cf. A000203 (sigma), A240976, A341748.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, Feb 18 2021