Mild Narcissistic Constants
The basic idea for this took shape after thinking about Mike Keith's wild narcissistic numbers. Can we come up with any criterion that evinces self-reference when numbers are expressed in simple-continued-fraction form? I had already concluded that Khinchin's constant was itself a kind of strong narcissistic constant, insofar as the geometric mean of its constituent parts is the number itself. Technically, I suppose, this remains to be proved.
In 1935, A.Ya. Khinchin showed that - for almost all irrationals - the frequency of occurrence of a given number 'n' in its (infinite) simple continued fraction expansion is (Log[(n + 1)^2] - Log[(n + 1)^2 - 1])/Log[2] ("Log" here being the "natural" logarithm, i.e., to the base 'e'). In fact, Mathematica shows Sum[Log[(n + 1)^2/((n + 1)^2 - 1)]/Log[2], {n, 1, Infinity}] == 1.
The frequencies are (approximately) {.415037, .169925, .0931094, .0588937, .040642, .0297473, .0227201, .0179219, .0144996, ...}. In continued fraction form (28 terms of each the first 120 MNCs):
| #\Term> | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |
| 1 | 0 | 2 | 2 | 2 | 3 | 1 | 5 | 2 | 23 | 2 | 2 | 1 | 1 | 55 | 1 | 4 | 3 | 1 | 1 | 15 | 1 | 9 | 2 | 5 | 7 | 1 | 1 | 4 |
| 2 | 0 | 5 | 1 | 7 | 1 | 2 | 4 | 11 | 1 | 2 | 2 | 3 | 27 | 1 | 9 | 1 | 1 | 3 | 7 | 1 | 19 | 1 | 10 | 3 | 1 | 3 | 2 | 17 |
| 3 | 0 | 10 | 1 | 2 | 1 | 5 | 1 | 1 | 6 | 1 | 10 | 1 | 1 | 3 | 1 | 10 | 1 | 1 | 2 | 5 | 2 | 1 | 25 | 4 | 1 | 1 | 1 | 3 |
| 4 | 0 | 16 | 1 | 48 | 2 | 1 | 1 | 1 | 4 | 1 | 3 | 1 | 4 | 1 | 6 | 3 | 2 | 1 | 2 | 15 | 2 | 11 | 4 | 1 | 1 | 8 | 2 | 1 |
| 5 | 0 | 24 | 1 | 1 | 1 | 1 | 7 | 4 | 18 | 2 | 1 | 1 | 48 | 1 | 5 | 1 | 3 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 |
| 6 | 0 | 33 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 20 | 3 | 1 | 1 | 2 | 1 | 24 | 1 | 50 | 2 | 4 | 1 | 2 | 1 | 40 | 1 | 1 | 1 |
| 7 | 0 | 44 | 71 | 1 | 3 | 12 | 4 | 1 | 1 | 2 | 4 | 1 | 2 | 1 | 2 | 5 | 1 | 1 | 2 | 15 | 1 | 17 | 1 | 2 | 204 | 1 | 1 | 4 |
| 8 | 0 | 55 | 1 | 3 | 1 | 16 | 12 | 3 | 197 | 4 | 6 | 2 | 2 | 31 | 1 | 7 | 3 | 25 | 1 | 3 | 23 | 2 | 1 | 5 | 1 | 1 | 1 | 1 |
| 9 | 0 | 68 | 1 | 29 | 1 | 4 | 1 | 7 | 4 | 1 | 1 | 1 | 39 | 1 | 2 | 104946 | 3 | 4 | 1 | 2 | 2 | 1 | 8 | 20 | 1 | 1 | 1 | 7 |
| 10 | 0 | 83 | 1 | 1 | 10 | 42 | 5 | 4 | 1 | 1 | 1 | 6 | 3 | 1 | 3 | 1 | 4 | 38 | 1 | 2 | 1 | 2 | 30 | 2 | 1 | 2 | 5 | 4 |
| 11 | 0 | 99 | 2 | 6 | 1 | 9 | 27 | 3 | 1 | 3 | 1 | 8 | 10 | 1 | 2 | 1 | 1 | 5 | 1 | 2 | 1 | 1 | 1 | 11 | 1 | 2 | 22 | 4 |
| 12 | 0 | 116 | 1 | 3 | 1 | 7 | 7 | 1 | 1 | 2 | 1 | 2 | 1 | 7 | 1 | 1 | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 1 | 16 | 5 | 1 | 6 |
| 13 | 0 | 135 | 1 | 1 | 24 | 1 | 1 | 4 | 9 | 2 | 1 | 2 | 1 | 2 | 25 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 8 | 40 | 2 | 11 | 1 | 9 |
| 14 | 0 | 155 | 1 | 1 | 1 | 1 | 2 | 1 | 17 | 18 | 1 | 2 | 3 | 2 | 2 | 1 | 173 | 9 | 16 | 1 | 5 | 4 | 3 | 2 | 9 | 2 | 1 | 1 |
| 15 | 0 | 177 | 10 | 8 | 1 | 4 | 2 | 7 | 11 | 3 | 3 | 1 | 1 | 5 | 4 | 2 | 1 | 5 | 1 | 3 | 1 | 6 | 1 | 20 | 2 | 1 | 8 | 1 |
| 16 | 0 | 199 | 1 | 35 | 1 | 2 | 2 | 11 | 2 | 24 | 1 | 5 | 1 | 3 | 1 | 4 | 9 | 3 | 4 | 19 | 1 | 1 | 2 | 1 | 1 | 8 | 2 | 8 |
| 17 | 0 | 224 | 4 | 3 | 2 | 2 | 2 | 1 | 5 | 2 | 2 | 1 | 6 | 1 | 2 | 1 | 12 | 2 | 2 | 31 | 1 | 2 | 1 | 1 | 48 | 1 | 1 | 1 |
| 18 | 0 | 249 | 1 | 7 | 3 | 2 | 2 | 1 | 17 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 14 | 1 | 2 | 2 | 2 | 2 | 1 | 6 | 1 | 6 | 3 |
| 19 | 0 | 276 | 1 | 10 | 2 | 1 | 1 | 1 | 1 | 5 | 90 | 1 | 1 | 2 | 2 | 2 | 3 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 3 | 17 | 1 | 11 |
| 20 | 0 | 305 | 3 | 51 | 1 | 3 | 1 | 11 | 39 | 32 | 1 | 3 | 7 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 6 | 3 | 2 | 1 | 1 | 1 | 1 | 1 |
| 21 | 0 | 335 | 7 | 3 | 11 | 4 | 10 | 78 | 1 | 30 | 1 | 1 | 1 | 4 | 2 | 1 | 46 | 2 | 3 | 1 | 1 | 1 | 3 | 1 | 2 | 1 | 2 | 6 |
| 22 | 0 | 366 | 3 | 21 | 4 | 1 | 3 | 2 | 4 | 9 | 4135 | 1 | 166 | 2 | 5 | 5 | 1 | 2 | 18 | 2 | 1 | 1 | 60 | 1 | 29 | 1 | 2 | 28 |
| 23 | 0 | 398 | 1 | 9 | 1 | 1 | 1 | 5 | 1 | 16 | 1 | 7 | 1 | 1 | 6 | 1 | 2 | 1 | 5 | 1 | 15 | 1 | 5 | 3 | 2 | 3 | 3 | 1 |
| 24 | 0 | 432 | 1 | 6 | 1 | 2 | 2 | 6 | 1 | 1 | 1 | 2 | 2 | 1 | 12 | 2 | 26 | 1 | 21 | 8 | 1 | 9 | 4 | 1 | 1 | 2 | 3 | 2 |
| 25 | 0 | 468 | 4 | 1 | 1 | 8 | 2 | 1 | 10 | 128 | 5 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 10 | 1 | 18 | 1 | 1 | 13 | 1 | 3 | 2 |
| 26 | 0 | 504 | 1 | 22 | 1 | 1 | 1 | 1 | 4 | 3 | 3 | 1 | 14 | 1 | 1 | 7 | 1 | 15 | 3 | 1 | 1 | 1 | 3 | 15 | 1 | 1 | 4 | 4 |
| 27 | 0 | 543 | 12 | 2 | 1 | 1 | 2 | 12 | 13 | 2 | 2 | 15 | 15 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 2 | 3 | 2 | 1 | 1 | 2 | 5 | 3 |
| 28 | 0 | 582 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 9 | 5 | 1 | 8 | 7 | 3 | 71 | 7 | 7 | 2 | 2 | 2 | 8 | 2 | 4 | 2 | 1 | 8 | 1 |
| 29 | 0 | 623 | 2 | 17 | 7 | 2 | 1 | 36 | 1 | 1 | 2 | 1 | 4 | 2 | 3 | 2 | 1 | 2 | 1 | 65 | 1 | 14 | 1 | 1 | 1 | 2 | 14 | 6 |
| 30 | 0 | 665 | 1 | 3 | 3 | 3 | 1 | 5 | 1 | 1 | 3 | 2 | 1 | 8 | 1 | 2 | 1 | 3 | 6 | 2 | 1 | 7 | 5 | 1 | 2 | 5 | 6 | 1 |
| 31 | 0 | 709 | 2 | 3 | 2 | 2 | 3 | 7 | 2 | 1 | 1 | 3 | 6 | 8 | 1 | 1 | 16 | 1 | 9 | 1 | 1 | 1 | 3 | 4 | 1 | 5 | 4 | 3 |
| 32 | 0 | 754 | 2 | 26 | 4 | 17 | 2 | 9 | 1 | 4 | 2 | 1 | 17 | 2 | 1 | 7 | 3 | 1 | 1 | 3 | 1 | 1 | 2 | 5 | 3 | 1 | 12 | 1 |
| 33 | 0 | 800 | 1 | 13 | 1 | 1 | 1 | 1 | 17 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 11 | 3 | 4 | 1 | 2 | 4 | 5 | 1 | 3 | 2 | 1 |
| 34 | 0 | 848 | 1 | 3 | 6 | 1 | 20 | 1 | 6 | 3 | 1 | 3 | 1 | 2 | 1 | 14 | 5 | 1 | 1 | 3 | 1 | 1 | 2 | 1 | 42 | 2 | 2 | 10 |
| 35 | 0 | 897 | 1 | 34 | 1 | 7 | 4 | 1 | 18 | 5 | 1 | 1 | 5 | 1 | 9 | 10 | 2 | 1 | 3 | 1 | 330 | 1 | 1 | 1 | 1 | 4 | 1 | 3 |
| 36 | 0 | 948 | 1 | 1 | 2 | 1 | 45 | 15 | 1 | 13 | 2 | 2 | 1 | 3 | 2 | 2 | 3 | 2 | 9 | 1 | 52 | 2 | 71 | 1 | 1 | 6 | 11 | 1 |
| 37 | 0 | 1000 | 1 | 1 | 3 | 1 | 4 | 2 | 4 | 1 | 56 | 1 | 1 | 2 | 1 | 34 | 3 | 1 | 1 | 2 | 12 | 1 | 5 | 1 | 2 | 51 | 1 | 1 |
| 38 | 0 | 1053 | 1 | 13 | 2 | 1 | 30 | 4 | 6 | 3 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 25 | 1 | 2 | 1 | 3 | 7 | 1 | 1 | 15 | 1 | 2 |
| 39 | 0 | 1108 | 1 | 2 | 4 | 1 | 2 | 50 | 1 | 1 | 2 | 1 | 4 | 12 | 8 | 33 | 26 | 5 | 36 | 1 | 5 | 2 | 2 | 9 | 3 | 1 | 1 | 2 |
| 40 | 0 | 1164 | 1 | 5 | 59 | 30 | 1 | 2 | 1 | 5 | 1 | 3 | 1 | 4 | 1 | 1 | 7 | 1 | 2 | 4 | 3 | 1 | 4 | 71 | 3 | 2 | 5 | 1 |
| 41 | 0 | 1222 | 2 | 1 | 2 | 1 | 5 | 4 | 12 | 3 | 5 | 5 | 16 | 1 | 89 | 1 | 1 | 5 | 1 | 20 | 20 | 4 | 2 | 1 | 1 | 10 | 1 | 9 |
| 42 | 0 | 1281 | 3 | 1 | 1 | 5 | 1 | 5 | 3 | 5 | 1 | 6 | 1 | 1 | 3 | 1 | 6 | 1 | 1 | 1 | 1 | 2 | 1 | 7 | 2 | 3 | 1 | 4 |
| 43 | 0 | 1341 | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 2 | 9 | 3 | 1 | 1 | 2 | 10 | 6 | 4 | 2 | 1 | 1 | 1 | 1 | 1 | 6 | 1 | 2 | 4 |
| 44 | 0 | 1403 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 4 | 2 | 2 | 3 | 5 | 13 | 1 | 103 | 1 | 2 | 1 | 10 | 1 | 2 | 13 | 2 |
| 45 | 0 | 1466 | 2 | 1 | 5 | 31 | 1 | 6 | 5 | 1 | 2 | 5 | 1 | 14 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 5 | 1 | 5 | 1 | 1 | 10 | 1 |
| 46 | 0 | 1530 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 7 | 3 | 4 | 1 | 1 | 1 | 2 | 2 | 3 | 1 | 1 | 1 | 10 | 3 | 2 | 1 | 1 | 24 | 1 |
| 47 | 0 | 1596 | 1 | 1 | 1 | 50 | 1 | 2 | 1 | 7 | 4 | 10 | 3 | 4 | 1 | 5 | 1 | 2 | 1 | 9 | 1 | 10 | 2 | 29 | 1 | 2 | 1 | 5 |
| 48 | 0 | 1663 | 1 | 8 | 1 | 45 | 6 | 2 | 2 | 1 | 3 | 32 | 1 | 2 | 33 | 2 | 1 | 8 | 1 | 1 | 2 | 8 | 6 | 13 | 1 | 1 | 1 | 3 |
| 49 | 0 | 1732 | 1 | 1 | 11 | 4 | 1 | 4 | 1 | 8 | 2 | 5 | 117 | 1 | 15 | 1 | 1 | 2 | 2 | 20 | 3 | 1 | 1 | 2 | 3 | 1 | 5 | 1 |
| 50 | 0 | 1802 | 1 | 1 | 8 | 18 | 191 | 1 | 5 | 2 | 1 | 50 | 4 | 2 | 1 | 7 | 1 | 3 | 25 | 1 | 7 | 1 | 17 | 1 | 21 | 1 | 3 | 1 |
| 51 | 0 | 1873 | 1 | 12 | 19 | 2 | 1 | 2 | 1 | 6 | 1 | 14 | 3 | 1 | 6 | 3 | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 4 | 4 | 1 | 9 |
| 52 | 0 | 1946 | 1 | 2 | 2 | 1 | 1 | 1 | 9 | 1 | 2 | 1 | 3 | 1 | 4 | 4 | 1 | 6 | 4 | 1 | 15 | 23 | 1 | 13 | 1 | 1 | 14 | 1 |
| 53 | 0 | 2020 | 1 | 6 | 1 | 2 | 1 | 1 | 1 | 43 | 1 | 1 | 1 | 10 | 1 | 1 | 19 | 6 | 2 | 4 | 26 | 1 | 16 | 11 | 1 | 2 | 1 | 2 |
| 54 | 0 | 2096 | 2 | 2 | 1 | 3 | 2 | 2 | 2 | 2 | 1 | 3 | 9 | 14 | 4 | 2 | 3 | 2 | 1 | 1 | 1 | 8 | 1 | 1 | 3 | 2 | 2 | 3 |
| 55 | 0 | 2173 | 2 | 1 | 3 | 12 | 16 | 1 | 2 | 1 | 14 | 1 | 7 | 5 | 1 | 2 | 1 | 9 | 3 | 12 | 1 | 1 | 4 | 4 | 7 | 1 | 6 | 1 |
| 56 | 0 | 2251 | 1 | 2 | 4 | 1 | 2 | 1 | 2 | 1 | 10 | 10 | 2 | 4 | 2 | 2 | 21 | 1 | 14 | 5 | 2 | 1 | 1 | 6 | 1 | 3 | 2 | 2 |
| 57 | 0 | 2331 | 2 | 2 | 75 | 1 | 5 | 2 | 1 | 3 | 1 | 2 | 9 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 12 | 1 | 2 | 1 | 1 | 2 |
| 58 | 0 | 2412 | 2 | 198 | 1 | 3 | 7 | 1 | 15 | 14 | 31 | 1 | 1 | 1 | 2 | 6 | 1 | 3 | 3 | 3 | 2 | 30 | 1 | 9 | 2 | 1 | 1 | 4 |
| 59 | 0 | 2494 | 1 | 58 | 1 | 2 | 1 | 4 | 1 | 2 | 4 | 1 | 16 | 5 | 3 | 1 | 117 | 1 | 22 | 4 | 1 | 1 | 1 | 3 | 1 | 6 | 2 | 4 |
| 60 | 0 | 2578 | 1 | 5 | 1 | 5 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 16 | 1 | 1 | 1 | 6 | 1 | 2 | 66 | 2 | 5 | 2 | 1 | 11 | 24 | 1 |
| 61 | 0 | 2664 | 8 | 1 | 197 | 7 | 11 | 3 | 3 | 2 | 1 | 1 | 1 | 14 | 1 | 1 | 1 | 1 | 25 | 2 | 5 | 2 | 2 | 25 | 5 | 1 | 10 | 6 |
| 62 | 0 | 2750 | 1 | 3 | 13 | 2 | 2 | 1 | 2 | 4 | 8 | 1 | 2 | 3 | 10 | 1 | 4 | 17 | 1 | 2 | 25 | 1 | 3 | 1 | 9 | 1 | 5 | 2 |
| 63 | 0 | 2838 | 1 | 3 | 1 | 1 | 1 | 2 | 1 | 6 | 1 | 79 | 1 | 2 | 7 | 1 | 10 | 1 | 1 | 1 | 1 | 2 | 5 | 1 | 6 | 1 | 112 | 4 |
| 64 | 0 | 2928 | 4 | 1 | 158 | 1 | 4 | 2 | 2 | 1 | 1 | 4 | 99 | 1 | 3 | 5 | 8 | 1 | 1 | 9 | 2 | 1 | 3 | 3 | 1 | 1 | 4 | 1 |
| 65 | 0 | 3019 | 394 | 1 | 299 | 12 | 4 | 1 | 5 | 1 | 1 | 2 | 3 | 3 | 11 | 1 | 2 | 5 | 1 | 54 | 2 | 1 | 1 | 5 | 1 | 1 | 7 | 1 |
| 66 | 0 | 3111 | 5 | 4 | 3 | 2 | 1 | 4 | 5 | 8 | 1 | 3 | 1 | 1 | 9 | 5 | 2 | 1 | 494 | 1 | 4 | 1 | 2 | 1 | 1 | 2 | 4 | 1 |
| 67 | 0 | 3204 | 1 | 3 | 3 | 1 | 1 | 1 | 22 | 1 | 2 | 1 | 1 | 2 | 7 | 4 | 2 | 1 | 2 | 2 | 12 | 1 | 61 | 1 | 1 | 1 | 293 | 13 |
| 68 | 0 | 3299 | 1 | 2 | 1 | 1 | 1 | 62 | 15 | 4 | 2 | 5 | 1 | 2 | 9 | 1 | 3 | 1 | 1 | 3 | 1 | 2 | 26 | 1 | 4 | 2 | 2 | 1 |
| 69 | 0 | 3396 | 13 | 2 | 2 | 7 | 1 | 2 | 3 | 3 | 5 | 22 | 1 | 4 | 4 | 1 | 10 | 1 | 1 | 1 | 3 | 4 | 3 | 1 | 1 | 1 | 5 | 1 |
| 70 | 0 | 3493 | 1 | 4 | 4 | 1 | 1 | 2 | 3 | 3 | 2 | 1 | 5 | 2 | 3 | 1 | 1 | 3 | 1 | 2 | 7 | 2 | 13 | 5 | 3 | 4 | 1 | 9 |
| 71 | 0 | 3592 | 1 | 12 | 1 | 28 | 1 | 2 | 2 | 4 | 1 | 4 | 1 | 5 | 4 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 11 | 1 | 15 | 1 | 6 |
| 72 | 0 | 3693 | 2 | 3 | 3 | 44 | 1 | 7 | 2 | 3 | 4 | 54 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 5 | 1 | 100 | 1 | 20 | 1 |
| 73 | 0 | 3795 | 3 | 18 | 3 | 7 | 1 | 4 | 1 | 2 | 1 | 7 | 18 | 2 | 1 | 8 | 1 | 2 | 19 | 78 | 5 | 1 | 4 | 8 | 1 | 5 | 2 | 1 |
| 74 | 0 | 3898 | 1 | 1 | 1 | 1 | 5 | 1 | 2 | 1 | 7 | 2 | 3 | 1 | 5 | 2 | 3 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 1 | 1 | 2 | 14 |
| 75 | 0 | 4003 | 3 | 1 | 2 | 6 | 1 | 1 | 4 | 1 | 5 | 1 | 1 | 1 | 1 | 1 | 2 | 15 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 |
| 76 | 0 | 4109 | 3 | 10 | 2 | 8 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 6 | 2 | 5 | 1 | 3 | 5 | 1 | 1 | 4 | 1 | 1 | 24 | 2 | 2 | 5 |
| 77 | 0 | 4216 | 1 | 3 | 5 | 1 | 1 | 76 | 1 | 1 | 1 | 2 | 2 | 2 | 19 | 1 | 1 | 1 | 1 | 3 | 1 | 2 | 1 | 10 | 1 | 1 | 14 | 13 |
| 78 | 0 | 4325 | 1 | 1 | 2 | 2 | 3 | 1 | 4 | 1 | 2 | 18 | 1 | 3 | 210 | 2 | 3 | 1 | 52 | 3 | 1 | 2 | 15 | 34 | 7 | 5 | 1 | 2 |
| 79 | 0 | 4435 | 1 | 3 | 1 | 7 | 1 | 5 | 2 | 4 | 13 | 1 | 1 | 32 | 1 | 62 | 4 | 1 | 3 | 3 | 1 | 5 | 4 | 1 | 2 | 2 | 2 | 1 |
| 80 | 0 | 4547 | 2 | 1 | 1 | 4 | 2 | 3 | 1 | 15 | 16 | 1 | 430 | 2 | 3 | 2 | 1 | 2 | 1 | 1 | 24 | 1 | 8 | 1 | 2 | 4 | 2 | 1 |
| 81 | 0 | 4660 | 2 | 1 | 1 | 1 | 260 | 4 | 1 | 3 | 14 | 1 | 9 | 1 | 1 | 66 | 1 | 3 | 3 | 11 | 6 | 1 | 28 | 3 | 1 | 259 | 1 | 2 |
| 82 | 0 | 4774 | 1 | 2 | 1 | 10 | 3 | 3 | 4 | 1 | 3 | 3 | 1 | 82 | 1 | 10 | 13 | 1 | 9 | 2 | 4 | 41 | 41 | 14 | 1 | 1 | 2 | 4 |
| 83 | 0 | 4890 | 2 | 3299 | 1 | 467 | 2 | 4 | 1 | 2 | 3 | 1 | 1 | 5 | 1 | 1 | 23 | 1 | 57 | 1 | 15 | 2 | 1 | 1 | 1 | 1 | 3 | 1 |
| 84 | 0 | 5007 | 1 | 1 | 1 | 3 | 1 | 4 | 32 | 198 | 1 | 2 | 22 | 4 | 29 | 1 | 1 | 2 | 24 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 1 |
| 85 | 0 | 5126 | 5 | 1 | 7 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 5 | 2 | 1 | 7 | 2 | 15 | 9 | 1 | 2 | 1 | 1 | 8 | 2 | 2 | 1 |
| 86 | 0 | 5246 | 11 | 1 | 5 | 2 | 2 | 1 | 4 | 1 | 1 | 4 | 5 | 17 | 2 | 2 | 5 | 3 | 4 | 3 | 1 | 7 | 1 | 1 | 1 | 1 | 2 | 4 |
| 87 | 0 | 5367 | 2 | 1 | 1 | 2 | 9 | 1 | 2575 | 5 | 16 | 1 | 1 | 22 | 2 | 1 | 33 | 3 | 1 | 6 | 9 | 1 | 1 | 92 | 11 | 1 | 4 | 636 |
| 88 | 0 | 5490 | 13 | 1 | 5 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 7 | 1 | 1 | 1 | 28 | 2 | 7 | 183 | 5 | 1 | 2 | 1 | 1 | 5 | 2 | 1 |
| 89 | 0 | 5614 | 6 | 1 | 6 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 22 | 12 | 48 | 1 | 3 | 4 | 4 | 2 | 1 | 1 | 1 | 2 |
| 90 | 0 | 5739 | 1 | 1 | 1 | 1 | 7 | 16 | 1 | 1 | 4 | 4 | 2 | 1 | 2 | 4 | 1 | 6 | 1 | 1 | 1 | 28 | 2 | 2 | 1 | 34 | 1 | 6 |
| 91 | 0 | 5866 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 57 | 2 | 3 | 1 | 3 | 2 | 3 | 3 | 11 | 1 | 3 | 3 | 3 | 1 | 2 |
| 92 | 0 | 5994 | 1 | 2 | 6 | 3 | 5 | 8 | 10 | 3 | 24 | 1 | 5 | 13 | 1 | 2 | 1 | 26 | 2 | 1 | 2 | 2 | 2 | 7 | 8 | 1 | 2483 | 3 |
| 93 | 0 | 6124 | 3 | 3 | 4 | 1 | 16 | 1 | 1 | 4 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 6 | 13 | 1 |
| 94 | 0 | 6255 | 3 | 3 | 1 | 5 | 2 | 1 | 8 | 1 | 2 | 1 | 4 | 2 | 4 | 1338 | 4 | 1 | 1 | 7 | 1 | 2 | 1 | 2 | 33 | 1 | 22 | 1 |
| 95 | 0 | 6387 | 1 | 2 | 3 | 4 | 3 | 9 | 38 | 1 | 10 | 1 | 1 | 1 | 3 | 1 | 3 | 1 | 1 | 4 | 11 | 2 | 1 | 1 | 5 | 2 | 1 | 2 |
| 96 | 0 | 6521 | 2 | 9 | 1 | 1 | 2 | 17 | 1 | 6 | 1 | 7 | 14 | 9 | 1 | 8 | 1 | 1 | 2 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| 97 | 0 | 6656 | 1 | 1 | 1 | 3 | 2 | 1 | 13 | 1 | 2 | 2 | 1 | 70 | 1 | 4 | 46 | 1 | 1 | 7 | 2 | 4 | 2 | 5 | 1 | 4 | 1 | 6 |
| 98 | 0 | 6793 | 5 | 3 | 2 | 2 | 2 | 6 | 1 | 7 | 1 | 7 | 1 | 1 | 10 | 1 | 2 | 52 | 1 | 5 | 1 | 1 | 3 | 16 | 1 | 17 | 1 | 1 |
| 99 | 0 | 6931 | 7 | 1 | 68 | 5 | 4 | 3 | 1 | 1 | 2 | 13 | 1 | 1 | 36 | 265 | 1 | 6 | 1 | 1 | 73 | 8 | 1 | 1 | 3 | 4 | 2 | 21 |
| 100 | 0 | 7070 | 2 | 4 | 3 | 2 | 4 | 6 | 23 | 1 | 4 | 67 | 1 | 2 | 1 | 28 | 1 | 1 | 1 | 18 | 1 | 1 | 1 | 1 | 1 | 20 | 2 | 5 |
| 101 | 0 | 7211 | 6 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 381 | 4 | 1 | 2 | 4 | 9 | 2 | 3 | 4 | 1 | 11 | 4 | 2 | 1 | 2 | 6 |
| 102 | 0 | 7353 | 3 | 1 | 32 | 1 | 6 | 5 | 1 | 1 | 4 | 1 | 2 | 3 | 2361 | 7 | 1 | 2 | 2 | 1 | 6 | 4 | 1 | 4 | 3 | 1 | 7 | 4 |
| 103 | 0 | 7496 | 1 | 2 | 1 | 2 | 1 | 605 | 1 | 3 | 2 | 2 | 2 | 4 | 2 | 1 | 9 | 333 | 33 | 2 | 2 | 2 | 1 | 2 | 1 | 3 | 1 | 2 |
| 104 | 0 | 7641 | 1 | 1 | 1 | 1 | 36 | 4 | 1 | 9 | 1 | 2 | 66 | 4 | 2 | 4 | 1 | 6 | 1 | 1 | 5 | 9 | 2 | 3 | 1 | 4 | 1 | 14 |
| 105 | 0 | 7787 | 1 | 5 | 1 | 9 | 2 | 1 | 11 | 9 | 1 | 1 | 3 | 1 | 11 | 2 | 6 | 1 | 2 | 1 | 23 | 2 | 4 | 1 | 1 | 1 | 4 | 118 |
| 106 | 0 | 7935 | 2 | 54 | 1 | 19 | 1 | 4 | 1 | 3 | 2 | 83 | 1 | 6 | 1 | 12 | 1 | 1 | 1 | 8 | 3 | 2 | 1 | 3 | 1 | 1 | 1 | 1 |
| 107 | 0 | 8084 | 1 | 1 | 10 | 1 | 3 | 1 | 5 | 1 | 17 | 1 | 1 | 1 | 1 | 10 | 7 | 38 | 1 | 3 | 12 | 3 | 2 | 1 | 3 | 2 | 3 | 2 |
| 108 | 0 | 8234 | 1 | 14 | 2 | 2 | 18 | 1 | 1 | 1 | 73 | 2 | 2 | 1 | 1 | 1 | 4 | 249 | 89 | 24 | 4 | 1 | 3 | 1 | 2 | 6 | 5 | 8 |
| 109 | 0 | 8386 | 1 | 2 | 1 | 3 | 4 | 3 | 3 | 13 | 5 | 1 | 47 | 2 | 1 | 1 | 2 | 1 | 3 | 3 | 3 | 5 | 1 | 6 | 1 | 5 | 2 | 1 |
| 110 | 0 | 8539 | 1 | 11 | 2 | 9 | 8 | 13 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 156 | 1 | 8 | 12 | 1 | 16 | 2 | 4 | 1 | 60 | 6 | 2 |
| 111 | 0 | 8694 | 2 | 29 | 2 | 5 | 2 | 1 | 58 | 2 | 7 | 15 | 1 | 2 | 1 | 1 | 2 | 6 | 2 | 1 | 3 | 9 | 3 | 6 | 1 | 5 | 840 | 27 |
| 112 | 0 | 8850 | 2 | 4 | 2 | 10 | 2 | 3 | 1 | 2 | 1 | 33 | 2 | 1 | 4 | 3 | 1 | 1 | 11 | 12 | 5 | 2 | 5 | 1 | 6 | 3 | 2 | 2 |
| 113 | 0 | 9007 | 1 | 3 | 1 | 6 | 13 | 1 | 1 | 1 | 1 | 2 | 1 | 23 | 2 | 1 | 4 | 5 | 4 | 1 | 9 | 3 | 3 | 3 | 3 | 1 | 2 | 10 |
| 114 | 0 | 9166 | 1 | 1 | 9 | 1 | 1 | 4 | 1 | 9 | 1 | 3 | 1 | 2 | 17 | 7 | 1 | 4 | 1 | 1 | 2 | 1 | 9 | 1 | 1 | 1 | 3 | 44 |
| 115 | 0 | 9326 | 1 | 1 | 1 | 3 | 1 | 4 | 2 | 5 | 7 | 20 | 3 | 1 | 2 | 3 | 1 | 4 | 6 | 1 | 7 | 2 | 10 | 15 | 2 | 2 | 4 | 2 |
| 116 | 0 | 9488 | 6 | 1 | 7 | 1 | 15 | 1 | 4 | 7 | 2 | 2 | 3 | 9 | 5 | 2 | 1 | 2 | 3 | 2 | 2 | 1 | 2 | 16 | 2 | 3 | 25 | 10 |
| 117 | 0 | 9651 | 28 | 1 | 3 | 3 | 1 | 8 | 1 | 1 | 3 | 2 | 4 | 1 | 3 | 1 | 2 | 3 | 10 | 1 | 4 | 1 | 1 | 2 | 1 | 1 | 5 | 3 |
| 118 | 0 | 9815 | 3 | 4 | 1 | 1 | 3 | 2 | 1 | 1 | 5 | 3 | 5 | 1 | 39 | 3 | 3 | 1 | 1 | 8 | 1 | 1 | 7 | 4 | 13 | 13 | 2 | 1 |
| 119 | 0 | 9980 | 1 | 35 | 1 | 3 | 1 | 7 | 2 | 1 | 6 | 1 | 3 | 1 | 35 | 32 | 2 | 1 | 1 | 1 | 11 | 1 | 31 | 20 | 2 | 1 | 1 | 12 |
| 120 | 0 | 10148 | 46 | 1 | 26 | 1 | 1 | 4 | 1 | 12 | 1 | 3 | 1 | 1 | 1 | 3 | 2 | 32 | 1 | 4 | 44 | 10 | 1 | 5 | 6 | 1 | 9 | 1 |
We expect, in the n-th mild narcissistic constant (MNC), the frequency of occurrence of the number 'n' to be equal to the constant itself!
Another way of looking at this is to consider term-1 of the constants: efop = {2, 5, 10, 16, 24, 33, 44, 55, 68, 83, 99, 116, 135, 155, 177, 199, 224, 249, 276, 305, 335, 366, 398, 432, 468, 504, 543, 582, 623, 665, 709, 754, 800, 848, 897, 948, 1000, 1053, 1108, 1164, 1222, 1281, 1341, 1403, 1466, 1530, 1596, 1663, 1732, 1802, 1873, 1946, 2020, 2096, 2173, 2251, 2331, 2412, 2494, 2578, 2664, 2750, 2838, 2928, 3019, 3111, 3204, 3299, 3396, 3493, 3592, 3693, 3795, 3898, 4003, 4109, 4216, 4325, 4435, 4547, 4660, 4774, 4890, 5007, 5126, 5246, 5367, 5490, 5614, 5739, 5866, 5994, 6124, 6255, 6387, 6521, 6656, 6793, 6931, 7070, 7211, 7353, 7496, 7641, 7787, 7935, 8084, 8234, 8386, 8539, 8694, 8850, 9007, 9166, 9326, 9488, 9651, 9815, 9980, 10148, ...}. As a first approximation, the inverse of each of these terms is equal to its MNC. The values represent then, if you will, the expected first-occurrence position-numbers of 'n' in their associated MNCs.
Mathematics, alas, is a tad chaotic... Here are the actual first-occurrence position-numbers in the (first 120) MNCs: afop = {5, 5, 13, 8, 14, 66, 105, 168, 52, 4, 23, 176, 61, 159, 161, 42, 53, 455, 185, 159, 417, 112, 166, 32, 159, 291, 271, 1361, 1122, 451, 1215, 1403, 397, 628, 1786, 1101, 1322, 417, 61, 3472, 1290, 170, 2087, 76, 3237, 1868, 1991, 2179, 430, 11, 6111, 7588, 167, 4093, 7807, 1942, 2913, 118, 2072, 1280, 863, 1675, 2318, 8476, 6444, 3067, 658, 3741, 5389, 738, 223, 504, 90, 210, 1312, 3148, 1237, 3683, 801, 2800, 1543, 13, 7400, 12462, 17354, 536, 2785, 7952, 816, 6316, 13760, 2399, 13081, 5776, 10165, 11941, 3908, 3774, 2502, 14245, 2669, 11001, 3187, 115, 40882, 5438, 15443, 14002, 4357, 3307, 23838, 4686, 17736, 7993, 4159, 9537, 10804, 5995, 5968, 12633, ...}.
We may note that the expected position-number of 2 in the 2nd MNC (term-5) is 2. If we only look at position-numbers that are exact multiples of the n-th MNC's term-1, what is the smallest multiple that brings us to a position containing 'n'?
For n = 1, terms-2,-4,-6,-8,-10 are non-1, but term-12 is a 1, thus the answer is 6. For n = 2, as we have already noted, the answer is 1. For n = 3, terms-10,-20,-30,...,-100 are all non-3, but term-110 is a 3, so the answer is 11. Here is the set of solutions for the first few values of 'n': mult = {6, 1, 11, 28, 16, 2, 88, 8, 208, 61, 20, 48, 76, 661, 308, 206, 256, 142, 144, 570, 4, 1093, 634, 84, 676, 575, 734, 968, 949, 51, 511, 552, 1487, 962, 1258, 875, 2636, 326, 871, ...}. How many terms 'n' have we have encountered when we reach these values? The answer: numb = {3, 1, 10, 23, 16, 1, 79, 5, 233, 63, 25, 33, 70, 643, 314, 211, 229, 135, 164, 604, 4, 1040, 684, 87, 673, 570, 770, 975, 952, 44, 536, 542, 1532, 924, 1262, 872, 2591, 320, 841, ...}. [mult - numb = {3, 0, 1, 5, 0, 1, 9, 3, -25, -2, -5, 15, 6, 18, -6, -5, 27, 7, -20, -34, 0, 53, -50, -3, 3, 5, -36, -7, -3, 7, -25, 10, -45, 38, -4, 3, 45, 6, 30, ...}]
Statistically, it is reasonable to expect the numbers in mult to match those in efop, so, it is unlikely we will we ever see another 1. How about the first number that repeats an existing one? :-)
21 November 1999 © Rarebit Dreams