A111879 -id:A111879 - cp's OEIS frontend

A111881 Numerators of row sums of array of ratios A111879(n,k)/A111880(n,k).

1, 1, 29, 1, 103, 253, 1669, 181, 30791, 849, 452993, 41003, 94949, 421117, 18358463, 446801, 124184839, 30064511, 80932487, 19812817, 211524139, 333707681, 4757207109, 2557825027, 105920383973, 14417396537, 4649180818987

Offset: 3

Wolfdieter Lang, Aug 23 2005

			The ratios a(n)/A069220(n) are: 1/2, 1/3, 29/12, 1/5, 103/20, 253/105, 1669/280, 181/63, 30791/2520, 849/385, ... (see the W. Lang link under A111879.)

Cf. A111879, A111880.

The denominators are given in A069220(n), n>=3.

a(n) = numerator(Sum_{k=1..phi(n)-1} r(n, k)), with phi(n) = A000010(n) (Euler's totient function) and the ratios r(n, k) = A111879(n, k)/A111880(n, k).

A069220 Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

Original entry on oeis.org

1, 1, 2, 3, 12, 5, 20, 105, 280, 63, 2520, 385, 27720, 6435, 8008, 45045, 720720, 85085, 4084080, 2909907, 3695120, 1322685, 5173168, 37182145, 118982864, 128707425, 2974571600, 717084225, 80313433200, 215656441, 2329089562800

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Apr 12 2002

Seiichi Manyama, Table of n, a(n) for n = 1..2310

Cf. A017666, A002805.
Cf. A093600 (numerator of this sum).
Also denominators of row sums of A111879/A111880. For the numerators see A111881.

Table[s=0; Do[If[GCD[i, n]==1, s=s+1/i], {i, n}]; Denominator[s], {n, 1, 35}]
Table[Denominator[Total[1/Select[Range[n],GCD[n,#]==1&]]],{n,40}] (* Harvey P. Dale, Jun 07 2020 *)

for(n=1,40,print1(denominator(sum(k=1,n,if(gcd(k,n)==1,1/k))),","))

G.f. A(x) (for fractions) satisfies: A(x) = -log(1 - x)/(1 - x) - Sum_{k>=2} A(x^k)/k. - Ilya Gutkovskiy, Mar 31 2020

More terms from Jason Earls, Apr 14 2002

A111880 Denominators of array which counts positive rational numbers (not including natural numbers).

Original entry on oeis.org

2, 3, 4, 3, 2, 5, 6, 5, 4, 3, 2, 7, 5, 3, 8, 7, 5, 4, 2, 9, 7, 3, 10, 9, 8, 7, 6, 5, 4, 3, 2, 11, 7, 5, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 13, 11, 9, 5, 3, 14, 13, 11, 8, 7, 4, 2, 15, 13, 11, 9, 7, 5, 3, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 17, 13, 11, 7, 5, 18, 17, 16, 15, 14, 13, 12

Offset: 3

Wolfdieter Lang, Aug 23 2005

Numerators are given by A111879.

The method to obtain the rationals r(n,k) for row n is described under A111879.

			Triangle begins:
  [2],
  [3],
  [4, 3, 2],
  [5],
  [6, 5, 4, 3, 2],
  [7, 5, 3],
  [8, 7, 5, 4, 2],
  [9, 7, 3],
  ...
The corresponding rationals are:
  [1/2],
  [1/3],
  [1/4, 2/3, 3/2],
  [1/5],
  [1/6, 2/5, 3/4, 4/3, 5/2],
  [1/7, 3/5, 5/3],
  [1/8, 2/7, 4/5, 5/4, 7/2],
  [1/9, 3/7, 7/3],
  ...

P. Dienes, The Taylor Series, Dover 1957, p. 8, eq.(1).

Wolfdieter Lang, Array of ratios and more.

Cf. A020652/A020653 if natural numbers are included.

Cf. A111879.

a(n, k) = denominator(r(n, k)), n>=3, k=1..phi(n)-1, with phi(n) = A000010(n) (Euler's totient function) and the ratios r(n, k) are defined for row n in A111879.

cp's OEIS Frontend

A111881 Numerators of row sums of array of ratios A111879(n,k)/A111880(n,k).

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A069220 Denominator of Sum_{1<=k<=n, gcd(k,n)=1} 1/k.

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A111880 Denominators of array which counts positive rational numbers (not including natural numbers).

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