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%I #76 Dec 23 2020 08:40:09
%S 5,242424,271070,9292071,29133316,70421305,215817165252,649661007154
%N 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.
%C 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 a(2) is not the first occurrence of 242424 in Pi (which is at position 242422) but the second. - _Hans Havermann_, Jul 26 2014
%C a(9) is greater than 5 * 10^13. - _Kang Seonghoon_, Nov 02 2020
%H Tom Crawford and Brady Haran, <a href="https://www.youtube.com/watch?v=W20aT14t8Pw">Strings and Loops within Pi</a>, Numberphile video (2020).
%H Google, <a href="https://storage.googleapis.com/pi50t/index.html">50 trillion digits of pi</a> (2020).
%e 5 is a term because 5 is the 5th digit of Pi (3.1415...).
%t 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 *)
%t dpi = RealDigits[Pi, 10, 10000010][[1]]; Select[Range[10000000], FromDigits[Take[dpi, {#, # - 1 + IntegerLength[#]}]] == # &] (* _Vaclav Kotesovec_, Feb 18 2020 *)
%Y Cf. A000796, A057680, A064810, A109514.
%K nonn,base,more
%O 1,1
%A Mike Keith (domnei(AT)aol.com), Oct 19 2000
%E a(4)-a(6) from _Colin Rose_, Mar 15 2006
%E a(7) from _Alan Eliasen_, May 10 2013
%E a(8) from _Alan Eliasen_, Jun 06 2013
%E Name clarified by _Kang Seonghoon_, Nov 02 2020
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