Array of positive rational numbers without natural numbers  A111879(n)/A111800(n), n=3..25.
 
 The row length is phi(n-1)= A000010(n-1) (Euler totient function).

  n/k     1     2     3      4      5       6      7      8      9     10     11     12     13     14    15    16     17    18    19   20     21

  3    1/2

  4    1/3
  
  5    1/4    2/3   3/2
  
  6    1/5

  7    1/6    2/5   3/4    4/3    5/2

  8    1/7    3/5   5/3

  9    1/8    2/7   4/5    5/4    7/2
 
 10    1/9    3/7   7/3

 11    1/10   2/9   3/8    4/7    5/6    6/5     7/4    8/3   9/2

 12    1/11   5/7   7/5

 13    1/12  2/11  3/10    4/9    5/8    6/7     7/6    8/5   9/4  10/3    11/2
 
 14    1/13  3/11  5/9     9/5   11/3

 15    1/14  2/13  4/11    7/8    8/7    11/4   13/2

 16    1/15  3/13  5/11    7/9    9/7    11/5   13/3

 17    1/16  2/15  3/14   4/13   5/12    6/11   7/10    8/9    9/8  10/7    11/6   12/5   13/4   14/3  15/2

 18    1/17  5/13  7/11   11/7   13/5

 19    1/18  2/17  3/16   4/15   5/14    6/13   7/12   8/11   9/10  10/9    11/8   12/7   13/6   14/5  15/4  16/3  17/2

 20    1/19  3/17  7/13   9/11   11/9    13/7   17/3

 21    1/20  2/19  4/17   5/16   8/13   10/11  11/10   13/8   16/5  17/4    19/2

 22    1/21  3/19  5/17   7/15   9/13    13/9   15/7   17/5   19/3

 23    1/22  2/21  3/20   4/19   5/18    6/17   7/16   8/15   9/14  10/13  11/12  12/11  13/10   14/9  15/8  16/7   17/6  18/5  19/4  20/3  21/2

 24    1/23  5/19  7/17  11/13  13/11    17/7   19/5

 25    1/24  2/23  3/22   4/21   6/19    7/18   8/17   9/16  11/14  12/13  13/12  14/11   16/9   17/8  18/7  19/6   21/4  22/3  23/2
 .
 .
 .
 
  n/k     1     2     3      4      5       6      7      8      9     10     11     12     13     14    15    16     17    18    19   20     21    

################################################################################################################################################


 The sequence of row length is [1, 1, 3, 1, 5, 3, 5, 3, 9, 3, 11, 5, 7, 7, 15, 5, 17, 7, 11, 9, 21, 7, 19, ...], n>=3, 

 which is A000010(n)-1, with Euler's totient function phi(n)=A000010(n). 

 

 The row sums give, for n=3.. 35: 
 
 [1/2, 1/3, 29/12, 1/5, 103/20, 253/105, 1669/280, 181/63, 30791/2520, 849/385, 452993/27720, 41003/6435, 94949/8008, 421117/45045, 18358463/720720, 446801/85085, 124184839/4084080, 30064511/2909907, 80932487/3695120, 19812817/1322685, 211524139/5173168, 333707681/37182145, 4757207109/118982864, 2557825027/128707425, 105920383973/2974571600, 14417396537/717084225, 4649180818987/80313433200, 1725933683/215656441, 148699793966557/2329089562800, 129873768313829/4512611027925, 3140321675333/69458178400, 8108563907819/265447707525, 4393669984649/75014832672]
 

 The numerators of the row sums are, for n=3..35: (see A111802).
 
 [1, 1, 29, 1, 103, 253, 1669, 181, 30791, 849, 452993, 41003, 94949, 421117, 18358463, 446801, 124184839, 30064511, 80932487, 19812817, 211524139, 333707681, 4757207109, 2557825027, 105920383973, 14417396537, 4649180818987, 1725933683, 148699793966557, 129873768313829, 3140321675333, 8108563907819, 4393669984649]

 The denominators of the row sums are, for n=3..35: (see  A069220(n))
 
 [2, 3, 12, 5, 20, 105, 280, 63, 2520, 385, 27720, 6435, 8008, 45045, 720720, 85085, 4084080, 2909907, 3695120, 1322685, 5173168, 37182145, 118982864, 128707425, 2974571600, 717084225, 80313433200, 215656441, 2329089562800, 4512611027925, 69458178400, 265447707525, 75014832672]


#######################################################################################################################################