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

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

Author

Wolfdieter Lang, Aug 23 2005

Keywords

• nonn
• easy
• frac
• tabf

Comments

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).

Examples

			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],
  ...
		

References

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

Links

• Wolfdieter Lang, Array of ratios and more.

Crossrefs

Row sums give A111881(n)/A069220(n), n>=3, see the W. Lang link.

Cf. A020652/A020653 if natural numbers are included.

Cf. A111880.

Formula

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.