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

Author

Mike Keith, Oct 19 2000

Keywords

• nonn
• base
• more

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

Examples

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

Links

• Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
• Google, 50 trillion digits of pi (2020).

Crossrefs

Cf. A000796, A057680, A064810, A109514.

Programs

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 *)

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