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PrimeFan's Listing of Esoteric Integer Sequences

Main | Digits of π

Relating to the Decimal Digits of π

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.

3141500001

Sequence

3, 31, 653, 4159, 14159

Description

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.


3141500002

Sequence

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

Description

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

Comments

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).


3141500003

Sequence

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

Description

Sum of the first n digits of π.

comments

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

Mathematica Command

Table[Sum[PiDigits50[[i]], {i, 1, n}], {n, 50}]

3141500004

Sequence

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

Description

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

comments

Mathematica Command

Select[Range[50], IntegerQ[#/PiDigits50[[#]]] &]

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


3141500005

Sequence

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

Description

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

comments

Mathematica Command

Select[Range[50], IntegerQ[#/PiDigits50[[#]]] &]

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


3141500006

Sequence

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

Description

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.

Comments

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.

Mathematica Command

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

Select[Range[49], IntegerQ[(%[[(# + 1)]] - 3)/#] &]]

3141500007

Sequence

1, 2, 9, 11, 16

Description

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.

Comments

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.

Mathematica Command

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

Select[Range[50], IntegerQ[%[[#]]/#] &]

3141500008

Sequence

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

Description

The nth digit of π to the nth power. The integer part 3 is considered to be the first digit, at position 1.

Comments

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!

Mathematica Command

Table[PiDigits50[[n]]^n, {n, 50}]

3141500009

Sequence

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

Description

The nth integer raised to the power of the nth digit of π. The integer part 3 is considered to be the first digit, at position 1.

Comments

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.

Mathematica Command

Table[n^PiDigits50[[n]], {n, 50}]

3141500047

Sequence

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

Description

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


3141501729

Sequence

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

Description

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


3141569105

Sequence

49979, 51560, 181137, 193159

Description

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