A057679 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 3 is the first digit.

5, 242424, 271070, 9292071, 29133316, 70421305, 215817165252, 649661007154

Offset

1,1

Comments

The average number of matches of length "n" digits is exactly 0.9. That is, we expect 0.9 matches with 1 digit, 0.9 matches with 2 digits, etc. Increasing the number of digits by a factor of 10 means that we expect to find 0.9 new matches. Increasing the search from 10^11 to 10^12 (which includes 10 times as much work) would thus only expect to find 0.9 new matches. - Alan Eliasen, May 01 2013 (corrected by Michael Beight, Mar 21 2020)

a(2) is not the first occurrence of 242424 in Pi (which is at position 242422) but the second. - Hans Havermann, Jul 26 2014

a(9) is greater than 5 * 10^13. - Kang Seonghoon, Nov 02 2020

Links

Table of n, a(n) for n=1..8.

Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).

Google, 50 trillion digits of pi (2020).

Example

5 is a term because 5 is the 5th digit of Pi (3.1415...).

Mathematica

StringsinPi[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi, 10, cc] // First, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}]; ] (* For example, StringsinPi[6] returns all 6-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
dpi = RealDigits[Pi, 10, 10000010][[1]]; Select[Range[10000000], FromDigits[Take[dpi, {#, # - 1 + IntegerLength[#]}]] == # &] (* Vaclav Kotesovec, Feb 18 2020 *)

Crossrefs

Cf. A000796, A057680, A064810, A109514.

Sequence in context: A243114 A038027 A237641 * A123751 A152516 A295532

Adjacent sequences: A057676 A057677 A057678 * A057680 A057681 A057682

Keyword

nonn,base,more

Author

Mike Keith (domnei(AT)aol.com), Oct 19 2000

Extensions

a(4)-a(6) from Colin Rose, Mar 15 2006

a(7) from Alan Eliasen, May 10 2013

a(8) from Alan Eliasen, Jun 06 2013

Name clarified by Kang Seonghoon, Nov 02 2020

Status

approved