

A057680


Selflocating strings within Pi: numbers n such that the string n is at position n (after the decimal point) in decimal digits of Pi.


19




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 BorelCantelli 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 randomlychosen real numbers in place of Pi have infinitely many terms.)  Charles R Greathouse IV, Apr 29 2013
a(10) is greater than 10^12.  Alan Eliasen, Jun 17 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 nearmiss '043611' occurs at position 43611.  S. Alwin Mao, Feb 18 2020


REFERENCES

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.


LINKS

Table of n, a(n) for n=1..9.
David G. Andersen, The PiSearch Page.
Tom Crawford and Brady Haran, Strings and Loops within Pi, Numberphile video (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 5digit members of the sequence.  Colin Rose, Mar 15 2006 *)
Do[If[RealDigits[Pi, 10, a=i+IntegerLength@i1, 1][[1, i;; a]]==IntegerDigits@i, Print@i], {i, 50000}] (* Giorgos Kalogeropoulos, Feb 21 2020 *)


CROSSREFS

Cf. A000796, A057679, A109513, A064810.
Sequence in context: A168665 A283027 A031829 * A157796 A186848 A211841
Adjacent sequences: A057677 A057678 A057679 * A057681 A057682 A057683


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


STATUS

approved



