A176341 -id:A176341 - cp's OEIS frontend

A050286 Starting position of the first occurrence of a string of at least n '7's in the decimal expansion of Pi.

13, 559, 1589, 1589, 162248, 399579, 3346228, 24658601, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308

Offset: 1

Eric W. Weisstein

a(10) > 2*10^9 according to the SubIdiom.com/pi search engine. - M. F. Hasler, Apr 13 2019
a(11) > 99*10^9. - Giovanni Resta, Oct 02 2019
a(15) > 22*10^12. - Dmitry Petukhov, Jan 27 2020
a(16) > 50*10^12. - Dmitry Petukhov, Oct 30 2021

Timothy Mullican, 50 trillion digits of pi
Peter Trüb, 22.4 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits.

Cf. A000796: Decimal expansion (or digits) of Pi.
First occurrence of n times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
First occurrence of exactly n times the same digit: A096755 (exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
Cf. A176341 (first occurrence of n).
Cf. A121280 = A068987 - 1 (first occurrence of concatenate(1,...,n)).

a(n) = min { A096761(k); k >= n }. - M. F. Hasler, Mar 19 2017

Edited by M. F. Hasler, Mar 19 2017
a(10) from Giovanni Resta, Oct 02 2019
a(11)-a(13) added by Dmitry Petukhov, Jan 13 2020
a(14) from Dmitry Petukhov, Jan 27 2020
a(15) from Dmitry Petukhov, Oct 30 2021

A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...

Original entry on oeis.org

7, 17, 22, 29, 34, 54, 64, 74, 77, 84, 90, 94, 103, 113, 115, 136, 137, 141, 150, 161, 166, 174, 186, 187, 204, 222, 230, 242, 245, 261, 276, 281, 290, 293, 299, 303, 327, 330, 334, 336, 338, 355, 375, 381, 407

Offset: 1

Simon Plouffe, Feb 20 2000

See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024

			Pi = 3.1415926... where the first '2' occurs as the 7th digit.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A037001 (= a(n) - 1: the same with different offset).
Cf. A053745 - A053753 (similar for digits 1 through 9).
Cf. A035117 (first occurrence of at least n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (first occurrence of exactly n '1's), A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763 (exactly n '9's), A050279 (exactly n '0's).
Cf. A121280 = A068987 - 1: position of "123...n" in Pi's decimals.
Cf. A176341: first occurrence of n in Pi's digits.
Cf. A088566 (primes in this sequence).

Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)

A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024

a(n) = A037001(n) + 1. - Georg Fischer, May 31 2021

Changed offset from 0 to 1 by Vincenzo Librandi, Oct 07 2013

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

A193940 Starting position of the first occurrence of the decimal number 10^n in the decimal expansion of Pi.

Original entry on oeis.org

1, 49, 854, 854, 387791, 2393355, 10359802, 13310435, 172330849, 2542542101, 95257866194, 1587461837108, 3186699229889, 3186699229889

Offset: 0

Kausthub Gudipati, Aug 10 2011

From Dmitry Petukhov, Jan 15 2020: (Start)
Digits 3,1,4,1,5,... are indexed 0,1,2,3,4,...
10^2 without '0' after it found at 1816 position, but a(3)=854 < 1816, therefore a(2)=a(3)=854.
10^12 without '0' after it found at 6029081077667 position, but a(13)=3186699229889 < 6029081077667, therefore a(12)=a(13). (End)

			Pi = 3.141592653589793238462643383279502884197169399375105.. The '1' (10^0) after the decimal point is at position 1. The '1' of the first occurrence of '10' (10^1) is at position 49.

Peter Trüb, 22.4 trillion digits of pi

Cf. A000796, A176341, A194351.

str = ToString[N[Pi-3, 2*10^8]]; s = "1"; Table[pos = StringPosition[str, s, 1][[1,1]] - 2; s = s <> "0"; pos, {9}] (* this code takes a long time. T. D. Noe, Aug 10 2011 *)

Edited by Hans Havermann, Jul 21 2014
a(9) from Hans Havermann, Jul 21 2014
a(10)-a(13) from Dmitry Petukhov, Jan 15 2020

A194351 Starting position of the first occurrence of a string of 2^n in the decimal expansion of Pi.

Original entry on oeis.org

1, 6, 2, 11, 40, 15, 22, 148, 1750, 1842, 12735, 26862, 27372, 2943, 37619, 39587, 106920, 820238, 76875, 47887, 6150809, 3660438, 17376657, 15416321, 162454456, 132295965, 265234498, 33844308, 4847933000, 671531549, 1122335995, 2894348872, 763748417

Offset: 0

Kausthub Gudipati, Aug 22 2011

a(46) > 50*10^12. - _Dmitry Petukhov, Oct 27 2021

			Pi = 3.141592653589793238462643383279502884197169399375105.. The '1' (2^0) after the decimal point is at position 1. The '1' of the first occurrence of '16' (2^4) is at position 40.

Dmitry Petukhov, Table of n, a(n) for n = 0..45
Timothy Mullican, 50 trillion digits of pi
Peter Trüb, 22.4 trillion digits of pi

Cf. A000796, A176341, A193940.

d = ToString[N[Pi-3, 1000000]]; Table[pos = StringPosition[d, ToString[2^n], 1]; If[pos == {}, Print["not enough digits for ", 2^n]; pos = 0, pos = pos[[1, 1]] - 2], {n, 0, 19}] (* T. D. Noe, Sep 02 2011 *)

a(n) = A032445(2^n)-1. - R. J. Mathar, Sep 02 2011

Terms corrected by D. S. McNeil, Sep 02 2011
a(29), a(32) from D. S. McNeil, Sep 03 2011
Edited by Hans Havermann, Jul 22 2014
a(28), a(30)-a(31) from Hans Havermann, Jul 22 2014
a(33)-a(43), a(45) from Dmitry Petukhov, Jan 27 2020
a(44) from Dmitry Petukhov, Oct 27 2021

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 *)

A281092 Position of the first occurrence of n in the decimal expansion of e.

Original entry on oeis.org

13, 2, 0, 17, 10, 11, 20, 1, 3, 12, 195, 200, 370, 27, 223, 201, 94, 88, 2, 108, 111, 87, 252, 16, 33, 92, 30, 0, 4, 131, 71, 189, 110, 142, 143, 17, 19, 270, 85, 106, 66, 124, 97, 134, 239, 10, 103, 25, 228, 34, 235, 93, 15, 18, 76, 301, 153, 38, 325, 11, 20, 242, 32

Offset: 0

Bobby Jacobs, Jan 21 2017

The 2 before the decimal point is counted as position 0.

This differs from A078197(n) at n = 2, 27, 271, 2718, ... .

Cf. A001113, A051238, A078197, A088576, A176341.

With[{ed=RealDigits[E,10,500][[1]]},Flatten[Table[SequencePosition[ ed, IntegerDigits[n],1][[All,1]],{n,0,65}]]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 06 2017 *)

A332084 Triangle read by rows: T(n,k) is the smallest m >= 0 such that floor(Pi*n^m) == k (mod n), -1 if one does not exist, k = 0..n-1.

Original entry on oeis.org

0, 1, 0, 0, 2, 4, 1, 3, 2, 0, 1, 7, 3, 0, 8, 1, 9, 14, 0, 10, 2, 1, 7, 10, 0, 8, 6, 2, 3, 1, 8, 0, 9, 6, 14, 5, 10, 1, 2, 0, 3, 20, 18, 11, 5, 32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 5, 1, 8, 0, 13, 4, 2, 6, 9, 24, 12, 5, 1, 22, 0, 3, 17, 14, 18, 2, 6, 20, 10, 5, 1, 10, 0, 6, 9, 17, 14, 23, 7, 2, 21, 3

Offset: 1

Davis Smith, Aug 22 2020

Pi is normal in base n >= 2 if and only if in every row N, such that N is a power of n, -1 does not appear. Pi is absolutely normal if and only if -1 never appears.
Conjecture: Pi is absolutely normal, meaning that -1 will never appear.
This triangle is an instance of the more general f(n,k,r), where f(n,k,r) is the smallest m >= 0 such that floor(r*n^m) == k (mod n) (-1 if one does not exist) and r is irrational. The same conditions for normalcy apply.

			The triangle T(n,k) starts:
n\k   0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1:    0
2:    1   0
3:    0   2   4
4:    1   3   2   0
5:    1   7   3   0   8
6:    1   9  14   0  10   2
7:    1   7  10   0   8   6   2
8:    3   1   8   0   9   6  14   5
9:   10   1   2   0   3  20  18  11   5
10:  32   1   6   0   2   4   7  13  11   5
11:   5   1  22   0  13   4   2   6   9  24  12
12:   5   1  10   0   3  17  14  18   2   6  20  10
13:   5   1  10   0   6   9  17  14  23   7   2  21   3

Davis Smith, Rows n = 1..144 of triangle, flattened
David G. Anderson, The Pi-Search Page.
D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, An Empirical Approach to the Normality of π, Experimental Math., 21 (2012), 375-384.
C. Sevcik, Fractal analysis of Pi normality, arXiv:1608.00430 [math.GM], 2016.
P. Trueb, Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi, arXiv:1612.00489 [math.NT], 2016.
Wikipedia, Normal number.

Cf. A000796, A022844, A066643, A068425, A176341.

Positions of 0 through 9 in base 10: A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008.

A332084_row(n)={my(L=List(vector(n,z,-1)), m=-1); while(vecmin(Vec(L))==-1, my(Z=lift(Mod(floor(Pi*n^(m++)),n))+1); if(L[Z]<0,listput(L,m,Z))); Vec(L)}

T(n,3) = 0, n > 3.

A362058 The location of the first occurrence of n in the decimal expansion of phi (the golden ratio, 1.6180339887...).

Original entry on oeis.org

4, 0, 19, 5, 11, 22, 1, 10, 3, 7, 231, 34, 121, 55, 254, 366, 0, 35, 2, 188, 19, 54, 62, 131, 78, 213, 67, 63, 51, 174, 40, 137, 181, 5, 26, 56, 28, 98, 32, 6, 105, 90, 347, 27, 58, 21, 70, 102, 15, 11, 214, 394, 66, 111, 57, 768, 30, 48, 22, 166, 68, 1, 50

Offset: 0

James C. McMahon, Apr 06 2023

Locations in the expansion of phi are numbered 0 for the digit before the decimal point, 1 for the first digit after the decimal point, and so on.

			The first occurrence of 0 in phi occurs 4 places after the decimal point, so a(0)=4; 5 first occurs 22 places after the decimal point, so a(5)=22; 10 first occurs 231 places after the decimal point so a(10)=231.

Cf. A001622 (phi)
Cf. A088577 (1-based locations).
Cf. A078197 (for e), A176341 (for Pi), A014777 (for Pi but different indexing).

Table[-1 + SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 50}] &@ First@ RealDigits@ N[GoldenRatio, 10^4]

a(n) = A088577(n) - 1.

cp's OEIS Frontend

A050286 Starting position of the first occurrence of a string of at least n '7's in the decimal expansion of Pi.

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A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ...

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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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A193940 Starting position of the first occurrence of the decimal number 10^n in the decimal expansion of Pi.

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A194351 Starting position of the first occurrence of a string of 2^n in the decimal expansion of Pi.

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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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A281092 Position of the first occurrence of n in the decimal expansion of e.

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A332084 Triangle read by rows: T(n,k) is the smallest m >= 0 such that floor(Pi*n^m) == k (mod n), -1 if one does not exist, k = 0..n-1.

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