login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where 1 is the first digit. 21
1, 16470, 44899, 79873884, 711939213, 36541622473, 45677255610, 62644957128, 656430109694 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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)
Consequently, with the second Borel-Cantelli lemma, the expected number of terms in this sequence is infinite with probability 1. (Of course the sequence is not random, but almost all of the sequences corresponding to randomly-chosen real numbers in place of Pi have infinitely many terms.) - Charles R Greathouse IV, Apr 29 2013
a(1) & a(5) are the first occurrences in Pi of their respective strings; a(2) & a(4) are the second occurrences; a(3) is the fourth occurrence. - Hans Havermann, Jul 27 2014
A near-miss '043611' occurs at position 43611. - S. Alwin Mao, Feb 18 2020
a(10) > 5 * 10^13. - Kang Seonghoon, Nov 02 2020
Has no terms in common with A037008 (but see Mao comment above). - Charles R Greathouse IV, Jun 21 2022
REFERENCES
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
LINKS
David G. Andersen, The Pi-Search Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (2020).
Google, 50 trillion digits of pi (2020).
EXAMPLE
1 is a term because 1 is the first digit after the decimal point.
MATHEMATICA
StringsinPiAfterPoint[m_] := Module[{cc = 10^m + m, sol, aa}, sol = Partition[RealDigits[Pi, 10, cc] // First // Rest, m, 1]; Do[aa = FromDigits[sol[[i]]]; If[aa==i, Print[{i, aa}]], {i, Length[sol]}]; ] (* For example, StringsinPiAfterPoint[5] returns all 5-digit members of the sequence. - Colin Rose, Mar 15 2006 *)
Do[If[RealDigits[Pi, 10, a=i+IntegerLength@i-1, -1][[1, i;; a]]==IntegerDigits@i, Print@i], {i, 50000}] (* Giorgos Kalogeropoulos, Feb 21 2020 *)
CROSSREFS
Sequence in context: A168665 A283027 A031829 * A157796 A186848 A211841
KEYWORD
nonn,base,more
AUTHOR
Mike Keith (domnei(AT)aol.com), Oct 19 2000
EXTENSIONS
More terms from Colin Rose, Mar 15 2006
a(5) from Nathaniel Johnston, Nov 12 2010
a(6)-a(8) from Alan Eliasen, May 01 2013
a(9) from Alan Eliasen, Jun 06 2013
Name clarified by Kang Seonghoon, Nov 02 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)