p Statistics: Cuckoo Positions |
by JVSchmidt |
General |
It can be very interesting to explore the self referential qualities of a digit sequence.
One special kind of that is a correspondence between a subsequence and its absolute position in the whole sequence. If we find the sequence "klm...nop" exactly at position (klm...nop) we call this a COUCKOO POSITION. The position quasi predicts the sequence like the couckoo bird shouts its own name. The first Couckoo Position CP(1) is not hard to find because Pi starts with "1" at position 1 (after the decimal point): 14159265... But where comes the next Couckoo Position and how many of them will be found amongst our 4 billion digits? |
Serial Number k | CP(k) |
1 | 1 |
2 | 16.470 |
3 | 44.899 |
4 | 79.873.884 |
5 | 711.939.213 |
6 | ??? |
D | Number of positions (=hits) | Number of examin. series = 2*D + 1 | Number of hits per series |
0 | 5 | 1 | 5,00 |
1 | 19 | 3 | 6,33 |
2 | 33 | 5 | 6,60 |
3 | 49 | 7 | 7,00 |
4 | 72 | 9 | 8,00 |
5 | 84 | 11 | 7,64 |
6 | 97 | 13 | 7,46 |
7 | 112 | 15 | 7,47 |
8 | 123 | 17 | 7,23 |
9 | 143 | 19 | 7,53 |
10 | 153 | 21 | 7,29 |
Length of sequence L | sequence range | Number of hits |
1 (single digit) | 0-9 | 9 |
2 | 10-99 | 18 |
3 | 100-999 | 21 |
4 | 1.000-9.999 | 16 |
5 | 10.000-99.999 | 19 |
6 | 100.000-999.999 | 13 |
7 | 1.000.000-9.999.999 | 16 |
8 | 10.000.000-99.999.999 | 22 |
9 | 100.000.000-999.999.999 | 19 |