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A057680 Self-locating strings within Pi: numbers n such that the string n is at position n (after the decimal point) in decimal digits of Pi. 17


%S 1,16470,44899,79873884,711939213,36541622473,45677255610,62644957128,

%T 656430109694

%N Self-locating strings within Pi: numbers n such that the string n is at position n (after the decimal point) in decimal digits of Pi.

%C The total probability of finding a match 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

%C 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 a(10) is greater than 10^12. - _Alan Eliasen_, Jun 17 2013

%C 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

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

%H David G. Andersen, <a href="http://www.angio.net/pi/piquery">The Pi-Search Page</a>.

%e 1 is a term because 1 is the first digit after the decimal point.

%t 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

%Y Cf. A000796, A057679, A109513, A064810.

%K nonn,base,more

%O 1,2

%A Mike Keith (domnei(AT)aol.com), Oct 19 2000

%E More terms from _Colin Rose_, Mar 15 2006

%E a(5) from _Nathaniel Johnston_, Nov 12 2010

%E a(6)-a(8) from _Alan Eliasen_, May 01 2013

%E a(9) from _Alan Eliasen_, Jun 06 2013

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Last modified December 13 10:42 EST 2018. Contains 318086 sequences. (Running on oeis4.)