Research Article
Generalizing Benford's Law Using Power Laws: Application to Integer Sequences
Table 1
First digit distributions of some integer sequences.
| | Name of sequence | Sample size | Percentage of first digit occurrences |
| | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| | Benford law | | 30.1 | 17.6 | 12.5 | 9.7 | 7.9 | 6.7 | 5.8 | 5.1 | 4.6 | | Square | 100 | 21.0 | 14.0 | 12.0 | 12.0 | 9.0 | 9.0 | 8.0 | 7.0 | 8.0 | | Cube | 500 | 28.2 | 14.8 | 11.4 | 9.8 | 8.8 | 7.8 | 6.6 | 6.8 | 5.8 | | Cube | 1000 | 22.6 | 15.9 | 12.4 | 10.6 | 9.4 | 8.3 | 7.4 | 7.1 | 6.3 | | Cube | 10000 | 22.5 | 15.8 | 12.6 | 10.6 | 9.3 | 8.3 | 7.5 | 7.0 | 6.4 | | Square root | 99 | 19.2 | 17.2 | 15.2 | 13.1 | 11.1 | 9.1 | 7.1 | 5.1 | 3.0 | | Prime < 100 | 25 | 16.0 | 12.0 | 12.0 | 12.0 | 12.0 | 8.0 | 16.0 | 8.0 | 4.0 | | Prime < 1000 | 168 | 14.9 | 11.3 | 11.3 | 11.9 | 10.1 | 10.7 | 10.7 | 10.1 | 8.9 | | Prime < 10000 | 1229 | 13.0 | 11.9 | 11.3 | 11.3 | 10.7 | 11.0 | 10.2 | 10.3 | 10.3 | | Princeton number | 25 | 28.0 | 8.0 | 12.0 | 12.0 | 8.0 | 12.0 | 8.0 | 4.0 | 8.0 | | Mixing sequence | 618 | 28.3 | 14.6 | 11.5 | 9.9 | 7.6 | 7.8 | 8.1 | 6.6 | 5.7 | | Pentagonal number | 100 | 35.0 | 12.0 | 10.0 | 8.0 | 10.0 | 6.0 | 8.0 | 5.0 | 6.0 | | Keith number | 71 | 32.4 | 14.1 | 14.1 | 7.0 | 4.2 | 7.0 | 12.7 | 2.8 | 5.6 | | Bell number | 100 | 31.0 | 15.0 | 10.0 | 12.0 | 10.0 | 8.0 | 5.0 | 6.0 | 3.0 | | Catalan number | 100 | 33.0 | 18.0 | 11.0 | 11.0 | 8.0 | 8.0 | 4.0 | 3.0 | 4.0 | | Lucky number | 45 | 42.2 | 17.8 | 8.9 | 4.4 | 2.2 | 6.7 | 8.9 | 2.2 | 6.7 | | Ulam number | 44 | 45.5 | 13.6 | 6.8 | 6.8 | 4.5 | 6.8 | 4.5 | 6.8 | 4.5 | | Numeri ideoni | 65 | 30.8 | 18.5 | 13.8 | 10.8 | 6.2 | 3.1 | 7.7 | 6.2 | 3.1 | | Fibonacci number | 100 | 30.0 | 18.0 | 13.0 | 9.0 | 8.0 | 6.0 | 5.0 | 7.0 | 4.0 | | Partition number | 94 | 28.7 | 17.0 | 14.9 | 9.6 | 7.4 | 6.4 | 7.4 | 5.3 | 3.2 |
|
|
