cp's OEIS Frontend

A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.

%I A057680 #92 Jul 29 2024 09:59:21
%S A057680 1,16470,44899,79873884,711939213,36541622473,45677255610,62644957128,
%T A057680 656430109694
%N A057680 Self-locating strings within Pi: numbers n such that the string n is at position n in the decimal digits of Pi, where the initial digit 3 is at position 0.
%C A057680 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)
%C A057680 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
%C A057680 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
%C A057680 A near-miss '043611' occurs at position 43611. - _S. Alwin Mao_, Feb 18 2020
%C A057680 a(10) > 5 * 10^13. - _Kang Seonghoon_, Nov 02 2020
%C A057680 Has no terms in common with A037008 (but see Mao comment above). - _Charles R Greathouse IV_, Jun 21 2022
%D A057680 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
%H A057680 David G. Andersen, <a href="http://www.angio.net/pi/piquery">The Pi-Search Page</a>.
%H A057680 Tom Crawford and Brady Haran, <a href="https://www.youtube.com/watch?v=W20aT14t8Pw">Strings and Loops within Pi</a>, Numberphile video (2020).
%H A057680 Google, <a href="https://storage.googleapis.com/pi50t/index.html">50 trillion digits of pi</a> (2020).
%e A057680 1 is a term because the string of digits '1' occurs as the 1st digit after the decimal point.
%e A057680 Similarly, 16470 is a term because the string of digits '16470' occurs starting at position 16470 (after the decimal point) in the digits of Pi (although it already occurs earlier at position 1602). - _M. F. Hasler_, Jul 29 2024
%t A057680 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 *)
%t A057680 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 *)
%o A057680 (PARI) A057680_row(r=5)={my(M=10^r, R=[]); for(n=M\10, M-1, localprec(n+r); Pi\10^(1-r-n)%M==n && !print1(n",") && R=concat(R,n));R} \\ prints & returns the r-digit terms. - _M. F. Hasler_, Jul 29 2024
%Y A057680 Cf. A000796 (decimal digits of Pi), A057679 (variant where position 1 refers to the initial digit 3), A064810 (variant where position 0 refers to the first digit after the decimal point), A109513 (variant using chunks of m digits).
%K A057680 nonn,base,more
%O A057680 1,2
%A A057680 _Mike Keith_, Oct 19 2000
%E A057680 More terms from _Colin Rose_, Mar 15 2006
%E A057680 a(5) from _Nathaniel Johnston_, Nov 12 2010
%E A057680 a(6)-a(8) from _Alan Eliasen_, May 01 2013
%E A057680 a(9) from _Alan Eliasen_, Jun 06 2013
%E A057680 Name clarified by _Kang Seonghoon_, Nov 02 2020
%E A057680 Edited by _M. F. Hasler_, Jul 29 2024