I have to confess to not having heard of "early bird numbers" and "punctual bird numbers" before, even though they are quite plentiful. OEIS A116700 explains the former:
A116700 | "Early bird" numbers: write the natural numbers in a string 12345678910111213.... Sequence gives numbers that occur in the string ahead of their natural place, sorted into increasing order. |
As the OEIS comments explain:
"12" appears at the start of the string, ahead of its position after "11", so is a member. So are 123, 23, 1234, 234, 34, ... and sorting these into increasing order we get 12, 21, 23, 31, ...
The initial members of the sequence are (permalink):
12, 21, 23, 31, 32, 34, 41, 42, 43, 45, 51, 52, 53, 54, 56, 61, 62, 63, 64, 65, 67, 71, 72, 73, 74, 75, 76, 78, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 110, 111, 112, 121, 122, 123, 131, 132, 141, 142, 151, 152, 161, 162, 171
The initial members of the sequence are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 40, 44, 46, 47, 48, 49, 50, 55, 57, 58, 59, 60, 66, 68, 69, 70, 77, 79, 80, 88, 90, 100, 102, 103, 104, 105, 106, 107, 108, 109, 113, 114
It can be seen that 12 is missing from the above list because it is the first member of OEIS A116700. These numbers have an asymptotic density of zero. It's interesting to explore runs of consecutive numbers. For example, returning the early bird numbers, the record runs of consecutive numbers are as follows (starting number on left and length of run on the right):
- 12 --> 1
- 31 --> 2
- 41 -->3
- 51 -->4
- 61 --> 5
- 71 --> 6
- 81 --> 7
- 91 --> 9
- 210 --> 14
- 310 --> 25
- 410 --> 36
- 510 --> 47
- 610 --> 58
- 710 --> 69
- 810 --> 80
- 901 --> 99
- 2100 --> 124
- 3100 --> 235
- 4100 --> 346
- 5100 --> 457
- 6100 --> 568
- 7100 --> 679
- 8100 --> 790
- 9091 --> 909
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