%I #20 Mar 07 2022 03:17:22

%S 7,5,5,3,6,6

%N 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 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 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 Dick Boland, <a href="http://www.imathination.org/kappa/kappa.pdf">An Analog of the Harmonic Numbers Over the Squarefree Integers</a>, 2008.

%F 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 0.755366...

%t << 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 Cf. A138316, A138317 (numerators and denominators of the squarefree totient analogs of the harmonic numbers).

%Y Cf. A138312 (Mertens's B_3 minus Euler's constant), A083343 (Mertens's B_3), A001620 (Euler's constant).

%Y Cf. A000010, A008683, A001008, A002805.

%K nonn,cons,more

%O 0,1

%A Dick Boland (abstract(AT)imathination.org), Mar 13 2008