A050282 -id:A050282 - cp's OEIS frontend

A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

6, 16, 21, 28, 33, 53, 63, 73, 76, 83, 89, 93, 102, 112, 114, 135, 136, 140, 149, 160, 165, 173, 185, 186, 203, 221, 229, 241, 244, 260, 275, 280, 289, 292, 298, 302, 326, 329, 333, 335, 337, 354, 374, 380, 406, 423, 435, 456, 462, 477, 479, 484, 485, 500

Offset: 1

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

The first few primes in this sequence are 53, 73, 83, 89, 149, 173, 229, 241, 337, 479, 571, 613, 661, 757, 829, 877, 911, 977, 991, ... - M. F. Hasler, Jul 28 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Digits.
OEIS index to sequences related to Pi.

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A053746 (= a(n) + 1: the same with different offset).
Cf. A037000, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008 (similar for digits 1, ..., 9 and 0).
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.

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 2] (* Robert G. Wilson v, Mar 07 2011 *)

A037001_upto(N=999, d=2)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

a(n) ~ 10*n if Pi is normal, as generally assumed. - M. F. Hasler, Jul 28 2024

A096763 Position of the first occurrence of exactly n consecutive '9's in a row in the decimal expansion of Pi.

Original entry on oeis.org

5, 44, 2949, 17988, 19446, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 10542036048450, 5758910552709

Offset: 1

Robert G. Wilson v, Jul 07 2004

David G. Andersen, The Pi-Search Page.
Yasumasa Kanada, Statistical Distribution Information, Home Page, Computer Centre, The University of Tokyo.
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]

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).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

a(10)-a(11) from Giovanni Resta, Sep 30 2019

a(12) from Yasumasa Kanada, 2002 and a(13)-a(14) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019

A035117 a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.

Original entry on oeis.org

1, 94, 153, 12700, 32788, 255945, 4657555, 159090113, 812432526, 3961184001, 15647738228, 1041032609981, 3907688331257, 68635742334547

Offset: 1

Leonardo Bitran (lbitran(AT)reuna.cl)

Presently identical to A096755, which is the first occurrences of exactly n 1's in the digits of Pi. Will differ as soon as there's some a(n) = a(n+1) and equivalently, A035117(n) > A035117(n+1). - M. F. Hasler, Mar 17 2017

David G. Andersen, The Pi-Search Page.
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]
Eric Weisstein's World of Mathematics, Pi Digits.
Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information

Cf. A000796 (decimal expansion (or digits) of Pi).
Cf. A035117 (this), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).
Cf. A096755 (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).

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007
Edited, after re-establishing A096755, by M. F. Hasler, Mar 17 2017
a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019
a(14) from Dmitry Petukhov, Sep 19 2022

A048940 Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.

Original entry on oeis.org

5, 44, 762, 762, 762, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 5758910552709, 5758910552709

Offset: 1

Eric W. Weisstein

a(10) > 11*10^9 - 1. - Eric W. Weisstein, Jul 20 2013
a(15) > 22*10^12. - Dmitry Petukhov, Jan 29 2020
Pi digits 3,1,4,... are indexed 0,1,2,... Note that this is different from other sequences such as A049522, A084073 which use indices 1,2,3,... For example, the position of the curious accumulation of six 9s is called 762 here; the same position is called 763 in A049522, A084073. - Jeppe Stig Nielsen, Aug 21 2017

David G. Andersen, The Pi-Search Page. (Yields, as of today, an incorrect result of 66780105 for the first occurrence of eight "9"s. - _M. F. Hasler_, Mar 19 2017)
Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]
Eric Weisstein's World of Mathematics, Feynman Point
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).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.
Cf. A049522, A084073.

Module[{m, nn = 7}, m = First@ RealDigits@ N[Pi, 10^nn]; Array[ SequencePosition[m, ConstantArray[9, #]][[1, 1]] - 1 &, nn]] (* Michael De Vlieger, Mar 20 2017 *)

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
Edited by M. F. Hasler, Mar 19 2017
a(10)-a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada Laboratory, 2002 and a(13)-a(14) from Shigeru Kondo, 2011 added by Dmitry Petukhov, Dec 23 2019

A050279 a(n) is the starting position of the first occurrence of a string of at least n '0's in the decimal expansion of Pi.

Original entry on oeis.org

32, 307, 601, 13390, 17534, 1699927, 3794572, 172330850, 2542542102, 8324296435, 371247087572, 1755524129973, 3186699229890, 6381820482331

Offset: 1

Eric W. Weisstein

At least up to a(10), also the starting position of the first occurrence of a string of exactly n '0's in the decimal expansion of Pi, cf. A096764. - M. F. Hasler, Mar 19 2017, edited Sep 03 2017

a(15) > 22*10^12. - Dmitry Petukhov, Jan 28 2020

Shigeru Kondo, calculation of Pi to 12.8 * 10^9 digits, using the program PiFast of Xavier Gourdon

David G. Andersen, The Pi-Search Page.
Peter Trüb, 22.4 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits

See A096764 for another version.
Cf. A000796: Decimal expansion (or digits) of Pi.
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), A096764 (exactly n '0's).
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 concatenate(1,...,n): A121280 = A068987 - 1.

More terms from Colin B. Martin (martinc(AT)ram.net.au), Nov 25 2001
Edited by N. J. A. Sloane at the suggestion of M. F. Hasler, Aug 24 2007
Edited by M. F. Hasler, Mar 19 2017
Definition modified by N. J. A. Sloane, Sep 03 2017
a(11)-a(14) added by Dmitry Petukhov, Jan 12 2020

A096761 Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.

Original entry on oeis.org

13, 559, 4575, 1589, 162248, 399579, 3346228, 82144203, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308

Offset: 1

Robert G. Wilson v, Jul 07 2004

Differs from A050286 from a(3) > a(4) on. - M. F. Hasler, Mar 18 2017
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

David G. Andersen, The Pi-Search Page.
Timothy Mullican, 50 trillion digits of pi
Peter Trüb, 22.4 trillion 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).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.
Cf. A000796 (decimal expansion (or digits) of Pi).

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

A096762 Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.

Original entry on oeis.org

11, 34, 4985, 4751, 213245, 222299, 4722613, 239798471, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705

Offset: 1

Robert G. Wilson v, Jul 07 2004

a(8) > 2*10^8, a(9) = 46663520, a(10) = 3040319543.

Differs from A050287 from a(3) > A050287(3) = A050287(4) = a(4) on. - M. F. Hasler, Mar 19 2017

David G. Andersen, The Pi-Search Page.
Peter Trüb, 22.4 trillion digits of pi

Cf. A000796: Decimal expansion (or digits) of Pi.
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).
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 concatenate(1,...,n): A121280 = A068987 - 1.

Edited by M. F. Hasler, Mar 19 2017
a(8) via SubIdiom.com/pi search engine from M. F. Hasler, Apr 13 2019
a(11)-a(13) added by Dmitry Petukhov, Dec 30 2019
a(14) from Dmitry Petukhov, Sep 20 2022

A096757 Index of first occurrence of exactly n consecutive threes in a row in the decimal expansion of Pi.

Original entry on oeis.org

9, 24, 1698, 66846, 28467, 1129019, 710100, 36488176, 2011485307, 4663739959, 60422218263, 1379574176590, 26258139334603

Offset: 1

Robert G. Wilson v, Jul 07 2004

Differs from A050282 as soon as a(4) > a(5), while A050282(4) = A050282(5) = a(5). - M. F. Hasler, Apr 13 2019
a(12) > 99*10^9. - Giovanni Resta, Oct 02 2019
a(13) > 22*10^12. - Dmitry Petukhov, Jan 26 2020
a(14) > 50*10^12. - Dmitry Petukhov, Oct 28 2021

David G. Andersen, The Pi-Search Page.
Timothy Mullican, 50 trillion digits of pi
Peter Trüb, 22.4 trillion digits of pi

Cf. A050282 (same for at least three 3s); A050279, A035117, A096756, A096758, A096759, A096760, A096761, A096762, A096763 (analog for digits 0, 1, 2, 4, ...).

a(11) from Giovanni Resta, Oct 02 2019
a(12) from Dmitry Petukhov, Jan 26 2020
a(13) from Dmitry Petukhov, Oct 28 2021

A068987 a(n) is the first position in the digit sequence 3,1,4,1,5,9,.... of Pi where the pattern "1,2,...,n" occurs (where position of the initial 3 is 1).

Original entry on oeis.org

2, 149, 1925, 13808, 49703, 2458886, 9470345, 186557267, 523551503, 191278379840, 4368196101672

Offset: 1

Joseph L. Pe, Apr 01 2002

1. We may never know if a(n) is defined for all n.
2. We split up the digits of any number > 9 in the pattern, e.g., if n = 11, we search for the pattern "1,2,3,4,5,6,7,8,9,1,0,1,1".
3. The pattern "1,2,3,4,5,6" does not occur before the 100,000th term in the digit sequence of Pi.
Two more terms a(6) and a(7) were found via the referenced Pi-Search link [Andersen], through which 100 million digits of Pi are currently available. - Rick L. Shepherd, Oct 10 2002
200 million digits now available at Pi-Search page. - Rick L. Shepherd, Aug 06 2006
This sequence uses position = 1 for the initial digit 3 of Pi, while A121280(n) = a(n)-1 starts counting at 0, as does the "Pi search page" and sequences A035117, A050279 - A050287, A048940, A096755 - A096763. - M. F. Hasler, Mar 18 2017
a(10) > 2*10^9. - M. F. Hasler, Apr 13 2019
a(12) > 22*10^12. - Dmitry Petukhov, Jan 29 2020

Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 32.

D. G. Andersen, The Pi-Search Page
SubIdiom.com, Irrational numbers search engine: π = 3.14159.... (Search within 2*10^9 digits, since at least 2009, maybe 2002.)
Peter Trüb, 22.4 trillion 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).
First occurrence of n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.
Cf. A000796: Decimal expansion (or digits) of Pi.

p = ToString[N[Pi, 50000]/10]; t = {1, 12, 123, 1234, 12345}; g[n_] := StringPosition[p, ToString[n]][[1]][[1]] - 2; Table[g[t[[i]]], {i, 1, 5}]

a(n) = A121280(n) + 1. - M. F. Hasler, Apr 13 2019

More terms from Rick L. Shepherd, Oct 10 2002
a(8) from Rick L. Shepherd, Aug 06 2006
Additional term a(9), using subidiom search engine, from M. F. Hasler, Apr 13 2019
a(10)-a(11) from Dmitry Petukhov, Jan 16 2020

A121280 Position where concatenate(1,...,n) occurs for the first time in the decimals of Pi (where 3, 1, 4,... are at position 0, 1, 2,...).

Original entry on oeis.org

1, 148, 1924, 13807, 49702, 2458885, 9470344, 186557266, 523551502, 191278379839, 4368196101671

Offset: 1

Alexander R. Povolotsky, Nov 03 2007

This sequence uses the same convention for the "position" as sequences A035117, A050279 - A050287, A048940, A096755 - A096763, while A068987(n) = a(n)+1 counts the positions of 3,1,4,.... as 1,2,3,... - M. F. Hasler, Mar 18 2017
a(10) > 2*10^9. - M. F. Hasler, Apr 13 2019
a(12) > 22*10^12. - Dmitry Petukhov, Jan 29 2020

Dave G. Andersen, The Pi-Search Page
Subidiom.com, Irrational numbers search engine: π = 3.14159...
Peter Trüb, 22.4 trillion 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; A121280 = A068987 - 1: first occurrence of concatenate(1,...,n).
Cf. A000796: Decimal expansion (or digits) of Pi.

a(n) = A068987(n) - 1.

New definition and cross-references from M. F. Hasler, Mar 18 2017
Additional term a(9), using subidiom search engine, from M. F. Hasler, Apr 13 2019
a(10)-a(11) from Dmitry Petukhov, Jan 16 2020

cp's OEIS Frontend

A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A096763 Position of the first occurrence of exactly n consecutive '9's in a row in the decimal expansion of Pi.

Views

Author

Keywords

Links

Crossrefs

Extensions

A035117 a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A048940 Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A050279 a(n) is the starting position of the first occurrence of a string of at least n '0's in the decimal expansion of Pi.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A096761 Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A096762 Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A096757 Index of first occurrence of exactly n consecutive threes in a row in the decimal expansion of Pi.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A068987 a(n) is the first position in the digit sequence 3,1,4,1,5,9,.... of Pi where the pattern "1,2,...,n" occurs (where position of the initial 3 is 1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica