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

3,4

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).

References

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

Links

Table of n, a(n) for n=3..98.

W. Lang, Array of ratios and more.

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.

Example

[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],...

Crossrefs

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

Cf. A020652/A020653 if natural numbers are included.

Sequence in context: A327192 A351651 A157813 * A193280 A114732 A123338

Adjacent sequences: A111876 A111877 A111878 * A111880 A111881 A111882

Keyword

nonn,easy,frac,tabf

Author

Wolfdieter Lang, Aug 23 2005

Status

approved