A111881 -id:A111881 - cp's OEIS frontend

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

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

A111879 Numerators of array which counts positive rational numbers (not including natural numbers).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 1, 2, 3, 4, 5, 1, 3, 5, 1, 2, 4, 5, 7, 1, 3, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 5, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 3, 5, 9, 11, 1, 2, 4, 7, 8, 11, 13, 1, 3, 5, 7, 9, 11, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 5, 7, 11, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12

Offset: 3

Wolfdieter Lang, Aug 23 2005

Denominators are given by A111880.
The sequence of row lengths is [1, 1, 3, 1, 5, 3, 5, 3, 9, 3, 11, 5, 7, 7, ...] = A000010(n)-1 = phi(n)-1, with Euler's totient function phi(n).
For n>=3 delete from the list [seq(j/n-j,j=1..n-2)] the natural numbers and the ratios p/q with (p,q) not 1 (p and q not relatively prime, i.e., p and q have a common divisor >1).

			Triangle begins:
  [1],
  [1],
  [1, 2, 3],
  [1],
  [1, 2, 3, 4, 5],
  [1, 3, 5],
  [1, 2, 4, 5, 7],
  [1, 3, 7],
  ...
The corresponding ratios 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.

Row sums give A111881(n)/A069220(n), n>=3, see the W. Lang link.
Cf. A020652/A020653 if natural numbers are included.
Cf. A111880.

a(n, k) = numerator(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) defined for row n above.

cp's OEIS Frontend

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

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