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A057680
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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.
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21
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
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LINKS
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EXAMPLE
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1 is a term because 1 is the first digit after the decimal point.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base,more,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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