A111879 - OEIS

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

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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,