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] #######################################################################################################################################