cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

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

Keywords

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' 3rd constant and gamma is Euler's constant.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Numerator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]], {n, 1, 60}]
  • PARI
    a(n) = numerator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

Formula

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

Views

Author

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

Keywords

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' 3rd constant and gamma is Euler's constant.

Examples

			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).

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]], {n, 1, 60}]
  • PARI
    a(n) = denominator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

Formula

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.

A138320 Numerators 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

0, 1, 2, 5, 7, 4, 173, 587, 1481, 1859, 20701, 18391, 241393, 275713, 148367, 548423, 2342059, 241321, 41436061, 19263077, 40604659, 43779103, 1009564739, 1907583043, 9002492327, 9603126977, 27322095131, 25887926681, 752184042199
Offset: 1

Views

Author

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

Keywords

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' 3rd constant and gamma is Euler's constant.

Examples

			Numerators 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)).

Links

Crossrefs

Cf. A138312, A138313, A138312, A138316, A138317, A138321, A083343, A001620.

Programs

  • Mathematica
    Table[Numerator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]
  • PARI
    for(n=1,60, print1(numerator(sum(k=1,n, moebius(k)^2/eulerphi(k)) - sum(j=1,n,1/j)), ", ")) \\ G. C. Greubel, Aug 31 2018

Formula

a(n) = Numerator[(Sum_{k=1..n} mu^2(k)/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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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