A138321 Denominators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n - H_n.

1, 2, 3, 12, 15, 5, 210, 840, 2520, 2520, 27720, 27720, 360360, 360360, 180180, 720720, 3063060, 340340, 58198140, 29099070, 58198140, 58198140, 1338557220, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600, 1164544781400, 582272390700, 18050444111700, 144403552893600

Offset

1,2

Comments

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.

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

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.

Example

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)), ...
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)).

Mathematica

Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]

Prog

(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

Crossrefs

Cf. A138312, A138313, A138312, A138316, A138317, A138320, A083343, A001620.

Sequence in context: A068603 A045878 A295398 * A015756 A163906 A302844

Adjacent sequences: A138318 A138319 A138320 * A138322 A138323 A138324

Keyword

frac,nonn

Author

Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008

Extensions

Data replaced by G. C. Greubel, Sep 14 2018

Status

approved