A032445 -id:A032445 - cp's OEIS frontend

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.

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

A066712 Inverse of A032445, with first appearance of string "0" in decimal expansion of Pi added to A032445.

Original entry on oeis.org

3, 1, 4, 15, 5, 9, 2, 6, 53, 35, 58, 8, 97, 7, 93, 32, 23, 38, 84, 46, 62, 264, 64, 43, 33, 383, 83, 327, 27, 795, 95, 50, 0, 28, 88, 841, 419, 19, 971, 71, 16, 69, 939, 39, 99, 937, 37, 75, 51, 10

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Jan 13 2002

First appearance of string n in decimal expansion of Pi

			a(4) = 15 because string 15 is the first appearance of a decimal string having first appearance starting at digit 4 of the decimal expansion of Pi.

Cf. A032445, A000796.

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

A032510 Scan decimal expansion of Pi until all n-digit strings have been seen; a(n) is last string seen.

Original entry on oeis.org

0, 68, 483, 6716, 33394, 569540, 1075656, 36432643, 172484538, 5918289042, 56377726040

Offset: 1

David W. Wilson

Fabrice Bellard, Link to data from Bellard's calculation of 2.7 trillion digits of pi
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 (number of digits in the decimal expansion of Pi that must be scanned to encounter all n-digit numbers).
Cf. A032445 (starting position of the first occurrence of n in the decimal expansion of Pi).

More terms from Michael Kleber

More terms from Fabrice Bellard (fabrice(AT)bellard.org)

A064467 Primes in Pi: a(n) = first position in decimal expansion of Pi that matches the n-th prime, or 0 if there is no such position.

Original entry on oeis.org

7, 1, 5, 14, 95, 111, 96, 38, 17, 187, 1, 47, 3, 24, 120, 9, 5, 220, 99, 40, 300, 14, 27, 12, 13, 853, 3487, 1488, 207, 363, 298, 1097, 860, 526, 2607, 394, 1658, 1411, 1183, 429, 439, 729, 1945, 169, 38, 705, 94, 136, 485, 186, 230, 1689, 1708, 1714, 1007, 614

Offset: 1

Reinhard Zumkeller, Oct 03 2001

			A000040(4) = 7 = A000796(14) and A000796(i) <> 7 for i < 14, so a(4) = 14.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

Cf. A000040, A000796, A032445.

pi = ToString[ N[ Pi, 4000]]; pi = StringDrop[pi, {2}]; Table[ StringPosition[pi, ToString[ Prime[ n]], 1][[1, 1]], {n, 60}]
With[{pidg=RealDigits[Pi,10,5000][[1]]},Table[SequencePosition[ pidg, IntegerDigits[ n]][[1,1]],{n,Prime[ Range[ 60]]}]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jun 01 2015 *)

A088576 Position of the first location of n in the decimal expansion of e.

Original entry on oeis.org

14, 3, 1, 18, 11, 12, 21, 2, 4, 13, 196, 201, 371, 28, 224, 202, 95, 89, 3, 109, 112, 88, 253, 17, 34, 93, 31, 1, 5, 132, 72, 190, 111, 143, 144, 18, 20, 271, 86, 107, 67, 125, 98, 135, 240, 11, 104, 26, 229, 35, 236, 94, 16, 19, 77, 302, 154, 39, 326, 12, 21, 243, 33

Offset: 0

Cino Hilliard, Nov 19 2003

Except for a(0), the same as A051238.

			The first 7 is in the 2nd position of the digits of e, so a(7) = 2.

T. D. Noe, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, e Digits
Eric Weisstein's World of Mathematics, Constant Digit Scanning

Cf. A001113 (decimal expansion of e).
Cf. A036900 (number of digits in the decimal expansion of e that must scanned to get all n-digit strings).
Cf. A032445 (positions in pi), A088577 (positions in phi).

Module[{nn=400,ed},ed=RealDigits[E,10,nn][[1]];Table[SequencePosition[ed,IntegerDigits[n],1][[1,1]],{n,0,70}]] (* Harvey P. Dale, Mar 30 2025 *)

trajedigitsd(n,m) = { default(realprecision,6000); p = exp(1)*10^5000; v = Vec(Str(p)); for(d=0,m, for(x=1,n, if(d<10, y = eval(v[x]), if(d<100, y = eval(v[x])*10 + eval(v[x+1]), if(d<1000, y = eval(v[x])*100 + eval(v[x+1])*10 + eval(v[x+2]), y = eval(v[x])*1000 + eval(v[x+1])*100 + eval(v[x+2])*10 + eval(v[x+3]) ); ); ); if(y == d,print1(x",");break); ); ) }

A232013 Number of iterations of A176341 ("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

4, 1, 12, 4, 13, 14, 11, 10

Offset: 0

M. F. Hasler, Nov 16 2013

See A232014 for a variant based on A032445 instead of A176341.
Some loops: (1), (711939213), (0, 32, 15, 3), (19, 37, 46), (40, 70, 96, 180, 3664, 24717, 15492, 84198, 65489, 3725, 16974, 41702, 3788, 5757, 1958, 14609, 62892, 44745, 9385, 169).
See Hans Havermann table (in links) for primary unknown-length evolutions.

			a(0)=4 since A176341(0)=32 (position of the first "0" in Pi's digits), A176341(32)=15 (position of the first "32" in Pi's digits), A176341(15)=3 (position of the first "15" in Pi's digits), A176341(3)=0 (position of the first "3" in Pi's digits); here we find the "0" again after 4 iterations, thus a(0)=4.
a(1)=1 since A176341(1)=1 (the first "1" occurs at position 1 in Pi's digits), which already "closes the loop" after 1 iteration.
a(2)=12 because the iterations yield 2 > 6 > 7 > 13 > 110 > 174 > 155 > 314 > 0 > 32 > 15 > 3 > 0, here we re-enter the loop (of length 4) after 12 iterations.

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.

pidigits = First[RealDigits[N[Pi, 10^6]]];
Table[ lst = {}; test = n; steps = 1;
While[AppendTo[lst, test]; !
   MemberQ[lst,
    test = First[
       First[SequencePosition[pidigits, IntegerDigits[test], 1]]] - 1],
steps++ ]; steps, {n, 0, 7}] (* Robert Price, Aug 31 2019 *)

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

Edited by Hans Havermann, Aug 01 2014

A035331 Base-1000 expansion of Pi.

Original entry on oeis.org

3, 141, 592, 653, 589, 793, 238, 462, 643, 383, 279, 502, 884, 197, 169, 399, 375, 105, 820, 974, 944, 592, 307, 816, 406, 286, 208, 998, 628, 34, 825, 342, 117, 67, 982, 148, 86, 513, 282, 306, 647, 93, 844, 609, 550, 582, 231, 725, 359, 408, 128, 481, 117, 450

Offset: 0

Jörg Zurkirchen

Start with a(0)=3; other terms are formed from triples of successive digits in the decimal expansion of Pi.

This sequence can be considered as a (pseudo)random generator with range 0..999. Its scatterplot graph is very similar to that of other random generators, e.g., A096558. - M. F. Hasler, May 14 2015

			Pi = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 ...

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A000796, A016062, A037244, A050819, A032445, A064467.

RealDigits[Pi,1000,60][[1]] (* Harvey P. Dale, Nov 22 2015 *)

default(realprecision,3*N=100);vector(N,i,Pi\1000^(1-i)%1000) \\ or: {P=Pi;vector(N,i,P\1+0*P=frac(P)*1000)} or {P=Pi/1000;vector(N,i,floor(P=frac(P)*1000))}. \\ M. F. Hasler, May 11 2015

a(n) = floor(Pi*10^(3n)) mod 1000. - M. F. Hasler, May 14 2015

More terms from Larry Reeves (larryr(AT)acm.org), Oct 04 2001

Better definition from Franklin T. Adams-Watters, Apr 10 2006

A051238 First appearance of string n in e.

Original entry on oeis.org

3, 1, 18, 11, 12, 21, 2, 4, 13, 196, 201, 371, 28, 224, 202, 95, 89, 3, 109, 112, 88, 253, 17, 34, 93, 31, 1, 5, 132, 72, 190, 111, 143, 144, 18, 20, 271, 86, 107, 67, 125, 98, 135, 240, 11, 104, 26, 229, 35, 236, 94, 16, 19, 77, 302, 154, 39, 326, 12, 21

Offset: 1

Harvey P. Dale

A088576 has the a(0) term.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, e Digits

Cf. A001113 (decimal expansion of e).
Cf. A088576 (same as a(n) but including the term for n = 0).
Cf. A032445.

(* Computing 200000 digits of e is sufficient up to n=10000 *) eString = RealDigits[E, 10, 200000] // First // ToString /@ # & // StringJoin; a[n_] := (p = StringPosition[eString, n // ToString, 1]; If[p == {}, 0, p[[1, 1]]]); Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 04 2013 *)
With[{eee=RealDigits[E,10,200000][[1]]},Transpose[Flatten[Table[ SequencePosition[ eee,IntegerDigits[n],1],{n,70}],1]][[1]]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Nov 20 2015 *)

A036903 Scan decimal expansion of Pi until all n-digit strings have been seen; a(n) is number of digits that must be scanned.

Original entry on oeis.org

32, 606, 8555, 99849, 1369564, 14118312, 166100506, 1816743912, 22445207406, 241641121048, 2512258603207

Offset: 1

Michael Kleber

Fabrice Bellard, Link to data from Bellard's calculation of 2.7 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits

Cf. A000796 (decimal expansion of Pi).
Cf. A080597 (= a(n) + 1).
Cf. A032445 (starting positions of the first occurrences of n the decimal expansion of Pi).
Cf. A032510 (last n-digit number seen when scanning for all n-digit numbers).
Cf. A036900-A036906.

a(n) = A080597(n) - 1.

a(8)-a(11) from fabrice(AT)bellard.org (Fabrice Bellard), Oct 23 2011

cp's OEIS Frontend

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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A066712 Inverse of A032445, with first appearance of string "0" in decimal expansion of Pi added to A032445.

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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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A032510 Scan decimal expansion of Pi until all n-digit strings have been seen; a(n) is last string seen.

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A064467 Primes in Pi: a(n) = first position in decimal expansion of Pi that matches the n-th prime, or 0 if there is no such position.

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A088576 Position of the first location of n in the decimal expansion of e.

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A232013 Number of iterations of A176341 ("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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A035331 Base-1000 expansion of Pi.

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