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

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

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy
• frac
• tabf

Comments

Numerators are given by A111879.

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

Examples

			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],
  ...
		

References

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

Links

• Wolfdieter Lang, Array of ratios and more.

Crossrefs

Cf. A020652/A020653 if natural numbers are included.

Cf. A111879.

Formula

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.