A138320 -id:A138320 - cp's OEIS frontend

A138316 Numerators of the squarefree totient analogs of the harmonic numbers F_n.

1, 2, 5, 5, 11, 13, 41, 41, 41, 11, 113, 113, 77, 241, 497, 497, 1009, 1009, 3067, 3067, 3127, 3199, 35549, 35549, 35549, 36209, 36209, 36209, 255443, 262373, 264221, 264221, 266993, 135229, 17048, 17048, 22859, 69347, 139849, 139849, 70271, 35713

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 27 2008

F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

			Numerators of F_n, e.g., F_1 = (1/1), F_2 = (1/1 + 1/1), ... F_11 = (1/1 + 1/1 + 1/2 + 0 + 1/4 + 1/2 + 1/6 + 0 + 0 + 1/4 + 1/10).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138317, A138320, A138321, A083343, A001620.

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

a(n) = numerator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

a(n) = numerator[sum(k=1 to n)mu^2(k)/phi(k)] where mu(k) is the Mobius function and phi(k) is Euler's Totient function.

A138317 Denominators of the squarefree totient analogs of the harmonic numbers F_n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 12, 12, 12, 3, 30, 30, 20, 60, 120, 120, 240, 240, 720, 720, 720, 720, 7920, 7920, 7920, 7920, 7920, 7920, 55440, 55440, 55440, 55440, 55440, 27720, 3465, 3465, 4620, 13860, 27720, 27720, 13860, 6930, 3465, 3465, 3465, 6930, 79695, 79695

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 27 2008

F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

			Denominators of F_n, e.g., - F_1 = (1/1), F_2 = (1/1 + 1/1), ... F_11 = (1/1 + 1/1 + 1/2 + 0 + 1/4 + 1/2 + 1/6 + 0 + 0 + 1/4 + 1/10).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138316, A138320, A138321, A083343, A001620.

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

a(n) = denominator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

a(n)=Denominator[sum(k=1 to n)mu^2(k)/phi(k)] where mu(k) is the Mobius function and phi(k) is Euler's Totient function.

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

Original entry on oeis.org

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

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

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

			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)).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

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

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

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

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.

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

cp's OEIS Frontend

A138316 Numerators of the squarefree totient analogs of the harmonic numbers F_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A138317 Denominators of the squarefree totient analogs of the harmonic numbers F_n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions