Factors of integers consisting only of the digit 9

9 = 32
99 = 32 × 11
999 = 33 × 37
9999 = 32 × 11 × 101
99999 = 32 × 41 × 271
999999 = 33 × >>>> 7 <<<< × 11 × 13 × 37
9 999999 = 32 × 239 × 4649
99 999999 = 32 × 11 × 73 × 101 × 137
999 999999 = 34 × 37 × 333667
9999 999999 = 32 × 11 × 41 × 271 × 9091
99999 999999 = 32 × 21649 × 513239
999999 999999 = 33 × 7 × 11 × 13 × 37 × 101 × 9901
9 999999 999999 = 32 × 53 × 79 × 265 371653
99 999999 999999 = 32 × 11 × 239 × 4649 × 909091
999 999999 999999 = 33 × 31 × 37 × 41 × 271 × 2 906161
9999 999999 999999 = 32 × 11 × >>>> 17 <<<< × 73 × 101 × 137 × 5 882353
99999 999999 999999 = 32 × 2 071723 × 5363 222357
999999 999999 999999 = 34 × 7 × 11 × 13 × >>>> 19 <<<< × 37 × 52579 × 333667
9 999999 999999 999999 = 32 × 1 111111 111111 111111
99 999999 999999 999999 = 32 × 11 × 41 × 101 × 271 × 3541 × 9091 × 27961
999 999999 999999 999999 = 33 × 37 × 43 × 239 × 1933 × 4649 × 10 838689
9999 999999 999999 999999 = 32 × 112 × >>>> 23 <<<< × 4093 × 8779 × 21649 × 513239
99999 999999 999999 999999 = 32 × 11111 111111 111111 111111
999999 999999 999999 999999 = 33 × 7 × 11 × 13 × 37 × 73 × 101 × 137 × 9901 × 99 990001
9 999999 999999 999999 999999 = 32 × 41 × 271 × 21401 × 25601 × 182521 213001
99 999999 999999 999999 999999 = 32 × 11 × 53 × 79 × 859 × 265 371653 × 1058 313049
999 999999 999999 999999 999999 = 35 × 37 × 757 × 333667 × 440 334654 777631
9999 999999 999999 999999 999999 = 32 × 11 × >>>> 29 <<<< × 101 × 239 × 281 × 4649 × 909091 × 121 499449
99999 999999 999999 999999 999999 = 32 × 3191 × 16763 × 43037 × 62003 × 77843 839397

The line numbers N containing the factorisation of (N-1) 9's which include N as a factor (flagged by >>>> N <<<< above),
correspond to numbers whose reciprocal is a repeating decimal of (N-1) digits.

This is because a repeating decimal 0.`abcdefg` can be written as abcdefg/9999999. That simplifies to 1/N if N is a common factor.