The Digits of Pi

Today's numbered day is 24566 and a check with the OEIS showed its connection to the digits of \(\pi\). Specifically, OEIS A083625 records the starting positions of strings of three 6's in the decimal expansion of \(\pi\). The first elements in the sequence are as shown below:

2440, 3151, 4000, 4435, 5403, 6840, 10163, 10335, 10591, 13594, 15888, 16109, 18504, 20231, 21880, 21881, 23057, 23511, 24566, 25948, 26212, 27703, 27841, 29666, 29868, 29869, 32427, 32428, 33363, 36353, 38132, 40370, 40650, 43523

Wolfram Mathworld has collected some interesting information about peculiarities in the digits of \(\pi\). For a start, OEIS A050285 lists the starting position of the first occurrence of a string of \(n\) 6's in the decimal expansion of \(\pi\), starting with \(n\)=1. The initial terms are 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, 386980412. It can be seen that \(n\)=3, corresponding to 666, occurs initially at position 2440. 6666 (\(n\)=4) occurs at position 21880 and this is reflected in OEIS A083625 which shows 666 at 21880 and 21881.

Many OEIS sequences relate to the digits of \(\pi\). Here are some of them:

• starting positions where 0123456789 occurs (OEIS A101815)
  
  
• starting positions where 9876543210 occurs (OEIS A101816)
  
  
• starting positions of the first occurrence of \(n\)=0, 1, 2, ... in the decimal expansion of \(\pi\) (including the initial 3 and counting it as the first digit) are 33, 2, 7, 1, 3, 5, 8, 14, ... (OEIS A032445)
  
  
• \(\pi\)-primes, i.e., \(\pi\)-constant primes occur at 2, 6, 38, 16208, 47577, 78073, ... (OEIS A060421)
  
  
• starting positions for repeating digits e.g. 6666 occurs at 21880

0 - A050279: 32, 307, 601, 13390, 17534, 1699927, ... 

1 - A035117: 1, 94, 153, 12700, 32788, 255945, ...

2 - A050281: 6, 135, 1735, 4902, 65260, 963024, ...

3 - A050282: 9, 24, 1698, 28467, 28467, 710100, ...

4 - A050283: 2, 59, 2707, 54525, 808650, 828499, ...

5 - A050284: 4, 130, 177, 24466, 24466, 244453, ...

6 - A050285: 7, 117, 2440, 21880, 48439, 252499, ...

7 - A050286: 13, 559, 1589, 1589, 162248, 399579, ... 

8 - A050287: 11, 34, 4751, 4751, 213245, 222299, ... 

9 - A048940: 5, 44, 762, 762, 762, 762, 1722776, ...

999999 occurs at position 762 and is known as the Feynman point.

on January 7th 2021