A232013 -id:A232013 - cp's OEIS frontend

A032445 Number the digits of the decimal expansion of Pi: 3 is the first, 1 is the second, 4 is the third and so on; a(n) gives the starting position of the first occurrence of n.

33, 2, 7, 1, 3, 5, 8, 14, 12, 6, 50, 95, 149, 111, 2, 4, 41, 96, 425, 38, 54, 94, 136, 17, 293, 90, 7, 29, 34, 187, 65, 1, 16, 25, 87, 10, 286, 47, 18, 44, 71, 3, 93, 24, 60, 61, 20, 120, 88, 58, 32, 49, 173, 9, 192, 131, 211, 405, 11, 5, 128, 220, 21, 313, 23, 8, 118, 99, 606

Offset: 0

Jeff Burch, Paul Simon (paulsimn(AT)microtec.net)

See A176341 for a variant counting positions starting with 0, and A232013 for a sequence based on iterations of A176341. - M. F. Hasler, Nov 16 2013

			a(10) = 50 because the first "10" in the decimal expansion of Pi occurs at digits 50 and 51: 31415926535897932384626433832795028841971693993751058209749445923...

T. D. Noe, Table of n, a(n) for n=0..9999
M. J. Halm, More Sequences, Mpossibilities 83, April 2003.
Eric Weisstein's World of Mathematics, Constant Digit Scanning
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796 (decimal expansion of Pi).
Cf. A080597 (terms from the decimal expansion of Pi which include every combination of n digits as consecutive subsequences).
Cf. A032510 (last string seen when scanning the decimal expansion of Pi until all n-digit strings have been seen).
Cf. A064467 (primes in Pi).

p = ToString[FromDigits[RealDigits[N[Pi, 10^4]][[1]]]]; Do[Print[StringPosition[p, ToString[n]][[1]][[1]]], {n, 1, 100}]
With[{pi=RealDigits[Pi,10,1000][[1]]},Transpose[Flatten[Table[ SequencePosition[ pi,IntegerDigits[n],1],{n,0,70}],1]][[1]]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Dec 01 2015 *)

A032445(n)=my(L=#Str(n)); n=Mod(n, 10^L); for(k=L-1, 9e9, Pi\.1^k-n||return(k+2-L)) \\ Make sure to use sufficient realprecision, e.g. via \p999. - M. F. Hasler, Nov 16 2013

a(n) = A176341(n)+1. - M. F. Hasler, Nov 16 2013

More terms from Simon Plouffe. Corrected by Michael Esposito and Michelle Vella (michael_esposito(AT)oz.sas.com).

More terms from Robert G. Wilson v, Oct 04 2001

A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

Original entry on oeis.org

32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 49, 94, 148, 110, 1, 3, 40, 95, 424, 37, 53, 93, 135, 16, 292, 89, 6, 28, 33, 186, 64, 0, 15, 24, 86, 9, 285, 46, 17, 43, 70, 2, 92, 23, 59, 60, 19, 119, 87, 57, 31, 48, 172, 8, 191, 130, 210, 404, 10, 4, 127, 219, 20, 312, 22, 7, 117, 98, 605, 41

Offset: 0

Daniel E. Loeb, Apr 15 2010

It is unknown whether Pi is a normal number. If it is (at least in base 10) then this sequence is well defined.

The numbers a(n) refer to the position of the initial digit of n in the decimal expansion of Pi, where "3" is at position a(3)=0, "1" is at position a(1)=1, etc. This is also the numbering scheme used on the "Pi search page" cited among the LINKS. See A232013 for a sequence based on iterations of this one. See A032445 for a variant of the present sequence, where numbering starts at one. - M. F. Hasler, Nov 16 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Dave Andersen, Search within first 200,000,000 digits of pi
Michael D. Huberty, Ko Hayashi & Chia Vang, First 100,000 digits of pi
Simon Plouffe, First 50,000,000 digits of pi

Cf. A000796, A011545, A014777, A032445, A232013.

p=ToString[FromDigits[RealDigits[N[Pi, 10^4]][[1]]]]; Do[Print[StringPosition[p, ToString[n]][[1]][[1]] - 1], {n, 0, 100}] (* Vincenzo Librandi, Apr 17 2017 *)
With[{pid=RealDigits[Pi,10,800][[1]]},Flatten[Table[ SequencePosition[ pid,IntegerDigits[n],1],{n,0,70}],1]][[All,1]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 27 2019 *)

A176341(n)=my(L=#Str(n));n=Mod(n,10^L);for(k=L-1,9e9,Pi\.1^k-n||return(k+1-L)) \\ Make sure to use sufficient realprecision, e.g. via \p999. - M. F. Hasler, Nov 16 2013

pi = "314159265358979323846264338327950288419716939937510582097494459230..."
[ pi.find(str(i)) for i in range(10000) ]

a(n) = A032445(n)-1. - M. F. Hasler, Nov 16 2013

a(n) = 0 if n is in A011545, otherwise a(n) = A014777(n). - Pontus von Brömssen, Aug 31 2024

A228412 Number of iterations of A176341 ("position of m in Pi") starting with n until a loop is reached.

Original entry on oeis.org

0, 0, 8, 0, 9, 10, 7, 6

Offset: 0

M. F. Hasler, Nov 16 2013

"A loop is reached" means that an element x is reached such that (A176341^k)(x) = x for some k>0.

			a(0)=a(1)=a(3)=0 since 0 and 3 are elements of the loop 0 -> 32 -> 15 -> 3 -> 0, and 1 is a fixed point (i.e., loop of length 1) of A176341.
a(2)=8 is the number of steps in 2 -> 6 -> 7 -> 13 -> 110 -> 174 -> 155 -> 314 -> 0, at which point the previously mentioned loop is reached.

J. Navarro, Les secrets du nombre Pi (Book review, in French).
Pi-Search Page, Loop Sequences within Pi.
Ady TZIDON, Loops in Pi.

Cf. A176341, A232013, A232014, A032445.

A228412(n)={my(u=0);for(i=1,9e9,u+=1<A176341(n))&&return(i-A232013(n)))}

A232014 Number of iterations of A032445 ("position of n in Pi") until a value is reached for the second time, when starting with n, or -1 if no value is repeated.

Original entry on oeis.org

16, 4, 3, 5, 6, 1

Offset: 0

M. F. Hasler, Nov 16 2013

See A232013 for a variant based on A176341 instead of A032445.
Some loops: (5), (271070), (9292071), (40, 71), (2, 7, 14), (296, 1060, 13737, 133453, 646539, 294342, 141273). - Hans Havermann, Jul 26 2014
See Hans Havermann table (in links) for primary unknown-length evolutions. - Hans Havermann, Aug 06 2014

			a(1)=4 since A032445(1)=2 (the first "1" occurs after the initial "3" as second digit in Pi), A032445(2)=7 (the first "2" occurs as 7th digit of Pi's decimal expansion), A032445(7)=14, A032445(14)=2, which "closes the loop" after 4 iterations. (The initial value does not need to be part of the loop.)

David G. Andersen, Loop Sequences within Pi, on The Pi-Search Page (Search 2*10^8 decimal digits of Pi).
Hans Havermann, Information table of n, a(n) for n=0..100.
Joaquin Navarro, Les secrets du nombre Pi (Book review, in French).
James Taylor, Irrational Numbers Search Engine (Search 2*10^9 decimal digits of Pi).
Ady Tzidon, Loops in Pi.

Cf. A032445.

A232014(n)={my(u=0);for(i=1,9e9,u+=1<A032445(n))&&return(i))}

Definition modified by N. J. A. Sloane, Jul 29 2014

A280532 a(1) = a(2) = 1, a(n) = A014777(a(n-1) + a(n-2)), n >= 3.

Original entry on oeis.org

1, 1, 6, 13, 37, 31, 605, 1411, 7174, 15567, 608953, 78903, 334535, 611552, 105928, 2557047, 2979162, 3263358, 6242520, 7825254, 37404834, 267494881, 639174488

Offset: 1

Anders Hellström, Jan 13 2017

Dave Andersen, Probability of Finding Strings In Pi
Anders Hellström, Sage program
James Taylor, Irrational Numbers Search Engine (Search 2*10^9 decimal digits of Pi)

Cf. A014777, A032445, A038100, A176341, A232013.

Module[{a = {1, 1}, s = First@ RealDigits[N[Pi, 10^7]]}, Do[AppendTo[a, -1 + SequencePosition[s, IntegerDigits[ a[[n - 1]] + a[[n - 2]] ]][[1, 1]]], {n, 3, 20}]; a] (* Michael De Vlieger, Jan 14 2017 *)

cp's OEIS Frontend

A032445 Number the digits of the decimal expansion of Pi: 3 is the first, 1 is the second, 4 is the third and so on; a(n) gives the starting position of the first occurrence of n.

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A176341 a(n) = the location of the first appearance of the decimal expansion of n in the decimal expansion of Pi.

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A228412 Number of iterations of A176341 ("position of m in Pi") starting with n until a loop is reached.

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A232014 Number of iterations of A032445 ("position of n in Pi") until a value is reached for the second time, when starting with n, or -1 if no value is repeated.

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A280532 a(1) = a(2) = 1, a(n) = A014777(a(n-1) + a(n-2)), n >= 3.

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