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.

7, 5, 5, 3, 6, 6

Offset: 0

Author

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

Keywords

• nonn
• cons
• more

Comments

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.

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).

Examples

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Links

• Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers, 2008.

Crossrefs

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

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

Cf. A000010, A008683, A001008, A002805.

Programs

Mathematica

<< 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]

Formula

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.