Array of positive rational numbers (including natural numbers) A038566(n)/A020653(n), n=1..24. 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 22 1 1/1 2 1/2 2/1 3 1/3 3/1 4 1/4 2/3 3/2 4/1 5 1/5 5/1 6 1/6 2/5 3/4 4/3 5/2 6/1 7 1/7 3/5 5/3 7/1 8 1/8 2/7 4/5 5/4 7/2 8/1 9 1/9 3/7 7/3 9/1 10 1/10 2/9 3/8 4/7 5/6 6/5 7/4 8/3 9/2 10/1 11 1/11 5/7 7/5 11/1 12 1/12 2/11 3/10 4/9 5/8 6/7 7/6 8/5 9/4 10/3 11/2 12/1 13 1/13 3/11 5/9 9/5 11/3 13/1 14 1/14 2/13 4/11 7/8 8/7 11/4 13/2 14/1 15 1/15 3/13 5/11 7/9 9/7 11/5 13/3 15/1 16 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 16/1 17 1/17 5/13 7/11 11/7 13/5 17/1 18 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 18/1 19 1/19 3/17 7/13 9/11 11/9 13/7 17/3 19/1 20 1/20 2/19 4/17 5/16 8/13 10/11 11/10 13/8 16/5 17/4 19/2 20/1 21 1/21 3/19 5/17 7/15 9/13 13/9 15/7 17/5 19/3 21/1 22 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 22/1 23 1/23 5/19 7/17 11/13 13/11 17/7 19/5 23/1 24 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 24/1 . . . n/k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ################################################################################################################################################ The sequence of row length is [1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,10,22,8,20,...] , n>=1, which is A000010(n+1), with Euler's totient function phi(n)=A000010(n). The row sums give, for n=1..35: See A111991(n)/ A069220(n), n>=1. [1, 5/2, 10/3, 77/12, 26/5, 223/20, 988/105, 3909/280, 748/63, 55991/2520, 5084/385, 785633/27720, 124658/6435, 207061/8008, 1096792/45045, 29889983/720720, 1893246/85085, 197698279/4084080, 85352744/2909907, 154834887/3695120, 47589202/1322685, 325333835/5173168, 1188897016/37182145, 7612795845/118982864, 5775510652/128707425, 183259245573/2974571600, 33778670612/717084225, 6897956948587/80313433200, 7979970472/215656441, 218572480850557/2329089562800, 269764710179504/4512611027925, 5362983384133/69458178400, 16868338256144/265447707525, 6944174295497/75014832672, 8830286876076/167133741775] The numerators of the row sums are, for n=1..35: (see A111991(n)). [1, 5, 10, 77, 26, 223, 988, 3909, 748, 55991, 5084, 785633, 124658, 207061, 1096792, 29889983, 1893246, 197698279, 85352744, 154834887, 47589202, 325333835, 1188897016, 7612795845, 5775510652, 183259245573, 33778670612, 6897956948587, 7979970472, 218572480850557, 269764710179504, 5362983384133, 16868338256144, 6944174295497, 8830286876076] The denominators of the row sums are, for n=1..35: (see A069220(n) ) [1, 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, 167133741775] #######################################################################################################################################