This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138313 #20 Mar 07 2022 03:17:22
%S A138313 7,5,5,3,6,6
%N A138313 Decimal expansion of constant 'kappa' = lim_{n -> infinity} (F_n - H_n), where H_n are harmonic numbers, F_n are squarefree totient analogs of H_n.
%C A138313 The squarefree totient analog of the harmonic number F_n is given by F_n = Sum_{k=1..n} mu^2(k)/phi(k) where mu(k) is the Möbius function and phi(k) is Euler's totient function.
%C A138313 Conjectured to be equivalent to Mertens's constant B_3 minus Euler's constant (A138312). B_3 - gamma is given by Sum_{i>=1} log p_i/(p_i*(p_i-1)), where p_i is the i^th prime = Sum_{j>=2} mu(j)*zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).
%H A138313 Dick Boland, <a href="http://www.imathination.org/kappa/kappa.pdf">An Analog of the Harmonic Numbers Over the Squarefree Integers</a>, 2008.
%F A138313 Limit_{n -> infinity} ((Sum_{k=1..n} mu^2(k)/phi(k)) - H_n), where mu(k) is the Möbius function, phi(k) is Euler's totient function and H_n is the n-th harmonic number.
%e A138313 0.755366...
%t A138313 << NumberTheory`NumberTheoryFunctions` prl = 100000; ts = 0; f = 1; While[f < 100000000000, If[SquareFreeQ[f], ts += N[1/EulerPhi[f], 15]; If[f > prl, Print[{f, ts, hn = N[HarmonicNumber[f], 15], N[ts - hn, 10]}]; prl += 100000]]; f += 1]
%Y A138313 Cf. A138316, A138317 (numerators and denominators of the squarefree totient analogs of the harmonic numbers).
%Y A138313 Cf. A138312 (Mertens's B_3 minus Euler's constant), A083343 (Mertens's B_3), A001620 (Euler's constant).
%Y A138313 Cf. A000010, A008683, A001008, A002805.
%K A138313 nonn,cons,more
%O A138313 0,1
%A A138313 Dick Boland (abstract(AT)imathination.org), Mar 13 2008