[seqfan] Superparticular trios

[Paulo Sachs]

Dear seqfan,
Do someone know someway to efficiently list all the numbers which are in superparticular proportion both with n and n+1?
For n = 1 to at least 24 there are http://oeis.org/A063123 (n) * 4 - 2 possibilities, but I don't know why.
One example: There are 30 numbers in superparticular proportion both with 14 and 15:
12, 105/8, 40/3, 27/2, 55/4, 180/13, 195/14, 225/16, 240/17, 85/6, 57/4, 100/7, 315/22, 72/5, 420/29, 29/2, 175/12, 44/3, 147/10, 280/19, 252/17, 119/8, 224/15, 196/13, 91/6, 168/11, 140/9, 63/4, 16 and 35/2.
Thanks,
P. Sachs