A278149 Triangle T(n, m) giving in row n the denominators of the fractions for the Farey dissection of order n.

2, 3, 3, 4, 5, 5, 4, 5, 7, 5, 5, 7, 5, 6, 9, 7, 8, 7, 7, 8, 7, 9, 6, 7, 11, 9, 7, 8, 7, 7, 8, 7, 9, 11, 7, 8, 13, 11, 9, 11, 10, 8, 12, 9, 9, 12, 8, 10, 11, 9, 11, 13, 8, 9, 15, 13, 11, 9, 11, 10, 11, 13, 12, 9, 9, 12, 13, 11, 10, 11, 9, 11, 13, 15, 9, 10, 17, 15, 13, 11, 14, 13, 11, 10, 11, 13, 12, 16, 11, 11, 16, 12, 13, 11, 10, 11, 13, 14, 11, 13, 15, 17, 10, 11, 19, 17, 15, 13, 11, 14, 13, 11, 17, 13, 11, 13, 12, 16, 11, 11, 16, 12, 13, 11, 13, 17, 11, 13, 14, 11, 13, 15, 17, 19, 11

Offset: 1

Author

Wolfdieter Lang, Nov 22 2016

Keywords

• nonn
• tabf
• frac
• easy

Comments

For the numerators see A278148.

The length of row n is A002088(n) = A005728(n) - 1.

See A278148 for the definition of the Farey dissection of order n of the interval [1/(n+1), n/(n+1)] into A015614(n) intervals J(n,j) = [l(n,j), r(n,j)] with r(n,j) = l(n,j+1), for j=1..A015614(n), where the fractions l(n,j) and r(n,j) are given in a comment of A278148 in terms of three consecutive members of the Farey fraction sequence of order n.

Examples

			The triangle T(n, m) begins:n\m 1  2  3  4  5  6  7  8  9 10 11 12 ...
1:  2
2:  3  3
3:  4  5  5  4
4:  5  7  5  5  7  5
5:  6  9  7  8  7  7  8  7  9  6
6:  7 11  9  7  8  7  7  8  7  9 11  7
...
n = 7: 8 13 11 9 11 10 8 12 9 9 12 8 10 11 9 11 13 8,
n = 8: 9 15 13 11 9 11 10 11 13 12 9 9 12 13 11 10 11 9 11 13 15 9,
n = 9: 10 17 15 13 11 14 13 11 10 11 13 12 16 11 11 16 12 13 11 10 11 13 14 11 13 15 17 10,
n = 10: 11 19 17 15 13 11 14 13 11 17 13 11 13 12 16 11 11 16 12 13 11 13 17 11 13 14 11 13 15 17 19 11.
........................................
For the fractions  A278148(n, m) / T(n,m) and the actual dissection intervals for n=5 see the examples for A278148.
		

References

• G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 121.
• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 29 - 31.

Crossrefs

Cf. A002088, A005728, A015614, A278148.

Formula

T(1, 1) = 2 and for n>= 2: T(n, 1) = n + 1, T(n, A002088(n)) = n + 1 and for

m = 2..(A002088(n) - 1): T(n, m) = denominator(l(n,m)) = denominator(p(n,m)/q(n,m) - 1/(q(n,m)*(q(n,m) + q(n,m-1)))).