A245677 Numerator of sum of fractions A182972(k) / A182973(k) such that A182972(k) + A182973(k) = n.

1, 1, 11, 1, 79, 26, 339, 34, 5297, 62, 69071, 1165, 11723, 9844, 471181, 2625, 8960447, 73244, 8231001, 243757, 1031626241, 151100, 4178462515, 2651758, 10396147563, 11843614, 64166447971, 362476, 1989542332021, 97275764008, 1830230212061, 57286319768

Offset: 3

Author

Reinhard Zumkeller, Jul 30 2014

Keywords

• nonn
• frac

Comments

A182972(n) and A182973(n) provide an enumeration of positive rationals < 1 arranged by increasing sum of numerator and denominator then by increasing numerator;

a(n) = numerator(sum(A182972(k)/A182973(k): k such that A182972(k)+A182973(k)=n));

A245718(n) = floor(a(n)/A245678(n)).

Examples

			.     |  (num, den) = (A182973, A182973) | num(sum)| den(sum)|   [sum]
.   n |  num/den,   num + den = n        | A245677 | A245678 | A245718
. ----+----------------------------------+---------+---------+--------
.   3 |  1/2                             |       1 |       2 |       0
.   4 |  1/3                             |       1 |       3 |       0
.   5 |  1/4, 2/3                        |      11 |      12 |       0
.   6 |  1/5                             |       1 |       5 |       0
.   7 |  1/6, 2/5, 3/4                   |      79 |      60 |       1
.   8 |  1/7, 3/5                        |      26 |      35 |       0
.   9 |  1/8, 2/7, 4/5                   |     339 |     280 |       1
.  10 |  1/9, 3/7                        |      34 |      63 |       0
.  11 |  1/10, 2/9, 3/8, 4/7, 5/6        |    5297 |    2520 |       2
.  12 |  1/11, 5/7                       |      62 |      77 |       0
.  13 |  1/12, 2/11, 3/10, 4/9, 5/8, 6/7 |   69071 |   27720 |       2
.  14 |  1/13, 3/11, 5/9                 |    1165 |    1287 |       0
.  15 |  1/14, 2/13, 4/11, 7/8           |   11723 |    8008 |       1
.  16 |  1/15, 3/13, 5/11, 7/9           |    9844 |    6435 |       1 .
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 3..1000
• Paul Yiu, Recreational Mathematics, 24.3.1 Appendix: Two enumerations of the rational numbers in (0,1), page 633.

Crossrefs

Cf. A245678 (denominator), A182972, A182973, A245718.

Programs

Haskell

import Data.Ratio ((%), numerator)
a245677 n = numerator $ sum
   [num % den | num <- [1 .. div n 2], let den = n - num, gcd num den == 1]