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%I #14 May 03 2024 19:52:29
%S 1,2,3,12,15,5,210,840,2520,2520,27720,27720,360360,360360,180180,
%T 720720,3063060,340340,58198140,29099070,58198140,58198140,1338557220,
%U 2677114440,13385572200,13385572200,40156716600,40156716600,1164544781400,582272390700,18050444111700,144403552893600
%N Denominators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n - H_n.
%C F_n - H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens's 3rd constant and gamma is Euler's constant.
%C Original data was given as {1, 1, 12, 24, 240, 80, 560, 3360, 30240, 7560, 831600, 831600, 93600, 21621600, 6177600, 12355200, 2940537600, 980179200, 55870214400, 2234808576, 3724680960, 177365760, 49597067520, 29758240512, 3719780064000} which is in error for this sequence. - _G. C. Greubel_, Sep 14 2018
%H G. C. Greubel, <a href="/A138321/b138321.txt">Table of n, a(n) for n = 1..1000</a>
%H Dick Boland, <a href="http://www.imathination.org/kappa/kappa.pdf">An Analog of the Harmonic Numbers Over the Squarefree Integers</a>
%F a(n) = denominator( (Sum_{k=1..n} mu(k)^2/phi(k)) - H_n) where mu(k) is the Mobius function, phi(k) is Euler's Totient function and H_n is the n-th Harmonic Number.
%e Denominators of F_n - H_n, e.g., -F_1 - H_1 = (1/1 - 1/1), F_2 = ((1/1 - 1/1) + (1/1 - 1/2)), ...
%e F_11 = ((1/1 - 1/1) + (1/1 - 1/2) + (1/2 - 1/3) + (0 - 1/4) + (1/4 - 1/5) + (1/2 - 1/6) + (1/6 - 1/7) + (0 - 1/8) + (0 - 1/9) + (1/4 - 1/10) + (1/10 - 1/11)).
%t Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]
%o (PARI) for(n=1, 60,print1(denominator(sum(k=1,n, moebius(k)^2/eulerphi(k) ) - sum(j=1, n, 1/j)), ", ")) \\ _G. C. Greubel_, Sep 14 2018
%Y Cf. A138312, A138313, A138312, A138316, A138317, A138320, A083343, A001620.
%K frac,nonn
%O 1,2
%A Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008
%E Data replaced by _G. C. Greubel_, Sep 14 2018