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?

Results

Couckoo Positions in the Pi-Sequence

Digits analyzed: 4 * 10 9
Analysis started at digit: 1

Serial Number k CP(k)
1 1
2 16.470
3 44.899
4 79.873.884
5 711.939.213
6 ???



Numbers of positions in the Pi-Sequence where [ value(position) - position ] <= D

Digits analyzed: 1 * 10 9
Analysis started at digit: 1

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



Dependence of the Numbers of hits in the Pi-Sequence where [ value(position) - position ] <= 10
from the Length of sequence L

Digits analyzed: 1 * 10 9
Analysis started at digit: 1

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


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