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