Even though for most practical purposes you only need π to 10 or so decimal places (heck, I've even done some "good enough" estimates with π as exactly 3), professional and amateur mathematicians alike study π to hundreds or thousands, sometimes millions, of decimal places. Will all this help you design a better tire or bake a better cake? Probably not. So for this reason, I consider integer sequences based on the decimal digits of π to be esoteric.

For our purposes here, we will consider the integer part of π, 3, to belong to the sequence of decimal digits of π (as listed in Sloane's A000796) unless otherwise noted. We will consider that initial 3 to be at position 0, unless otherwise noted.

3, 31, 653, 4159, 14159

First substring of n digits of π that spells a prime number. For example, for n = 3, the first substring of 3 digits, 314, is clearly not prime. Neither is 141, 415, and so on until we get to 653.

32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 49, 94

First instance of n within the digits of π. First n = 0.

This sequence was already in Sloane's OEIS, it's A014777. Though the integer part of 3 is not counted there, the results are almost exactly the same (there a(3) = 9). Also, a handy link to the Pi-Search Page is provided, saving me a lot of time poring over a printout of the digits of π. You can find just about anything in π. My Social Security number is in there! At least my credit card number is not in there (unless you split it into two parts, or if perhaps you search a few million digits more).

3, 4, 8, 9, 14, 23, 25, 31, 36, 39, 44, 52, 61, 68, 77, 80, 82, 85, 93, 97, 103, 105, 111, 115, 118, 121, 129, 132, 134, 141, 150, 155, 155, 157, 165, 173, 177, 178, 187, 194, 195, 201, 210, 213, 222, 231, 234, 241, 246, 247

Sum of the first n digits of π.

This one is also already in Sloane's OEIS, A046974.

1, 3, 6, 9, 10, 15, 16, 24, 27, 28, 36, 37, 40, 45, 49

Numbers n that are divisible by the n^{th} digit of π, with the integer part 3 considered to be the zeroeth digit.

Mathematica will complain about dividing by 0, but it should still give the desired results.

2, 4, 5, 14, 18, 20, 22, 24, 34, 38, 41, 42, 45, 50

Numbers n that are divisible by the n^{th} digit of π, with the integer part 3 considered to be the first digit.

Mathematica will complain about dividing by 0, but it should still give the desired results.

1, 3, 5, 7, 9, 13, 18, 20, 62

Numbers n that are divisible by the sum of the first n digits of π, with the integer part 3 considered to be the zeroeth digit and not part of the sum.

If you leave out the 62 and enter this sequence into the OEIS, it will respond with sequence A058871. This is probably nothing more than a tantalizing coincidence.

After executing the command to sum up the digits of π (see 3141500003), execute this command:

1, 2, 9, 11, 16

Numbers n that are divisible by the sum of the first n digits of π, with the integer part 3 considered to be the first digit.

Sloane's OEIS has me beat by three months on this one. See A098934. A major contributor to the OEIS, Robert G. Wilson, had Mathematica look at 2 million digits of π and came up with no more than I came up just by looking at the first 50. I will have my computer repeat his calculation overnight, and the next night I will see if I can find any more terms by looking at even more digits of π.

After executing the command to sum up the digits of π (see 3141500003), execute this command:

3, 1, 64, 1, 3125, 531441, 128, 1679616, 1953125, 59049, 48828125, 68719476736, 2541865828329, 678223072849, 205891132094649, 43046721, 131072, 387420489, 144115188075855872, 1099511627776, 21936950640377856, 4194304, 789730223053602816, 281474976710656, 847288609443, 2541865828329, 2417851639229258349412352, 22876792454961, 536870912, 22539340290692258087863249, 381520424476945831628649898809, 23283064365386962890625, 0, 17179869184, 40564819207303340847894502572032, 324518553658426726783156020576256, 18889465931478580854784, 1, 16423203268260658146231467800709255289, 6366805760909027985741435139224001, 1, 481229803398374426442198455156736, 107752636643058178097424660240453423951129, 984770902183611232881, 8727963568087712425891397479476727340041449, 78551672112789411833022577315290546060373041, 26588814358957503287787, 36703368217294125441230211032033660188801, 17763568394002504646778106689453125, 1

The n^{th} digit of π to the n^{th} power. The integer part 3 is considered to be the first digit, at position 1.

I think this sequence and 3141500009 very nicely illustrate the randomness of the digits of π. How easily in this one does a 0 or a 1 follow a very large number!

1, 2, 81, 4, 3125, 10077696, 49, 262144, 59049, 1000, 161051, 429981696, 10604499373, 105413504, 38443359375, 4096, 289, 5832, 16983563041, 160000, 85766121, 484, 148035889, 331776, 15625, 17576, 282429536481, 21952, 841, 21870000000, 26439622160671, 33554432, 1, 1156, 2251875390625, 2821109907456, 1874161, 38, 208728361158759, 163840000000, 41, 5489031744, 502592611936843, 85184, 756680642578125, 922190162669056, 103823, 587068342272, 282475249, 50

The n^{th} integer raised to the power of the n^{th} digit of π. The integer part 3 is considered to be the first digit, at position 1.

I think this sequence and 3141500008 very nicely illustrate the randomness of the digits of π. From n = 10 to 99, any 2-digit number in the sequence is equal to n.

119, 223, 537, 741, 896, 1093, 1221, 1263, 1268, 1274, 1320

Location of nth instance of the substring 47 within the digits of π.

8042, 11527, 19602, 20606, 26706, 59393, 84225, 93270, 93982, 109933

Location of nth instance of the substring 1729 within the digits of π.

49979, 51560, 181137, 193159

Location of nth instance of the substring 69105 within the digits of π.