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 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. 19
 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 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,changed 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

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Last modified July 4 12:21 EDT 2022. Contains 355075 sequences. (Running on oeis4.)