A096763 -id:A096763 - cp's OEIS frontend

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

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

A096758 Index of first occurrence of exactly n consecutive fours in a row in the decimal expansion of Pi.

Original entry on oeis.org

2, 59, 2707, 54525, 808650, 828499, 17893953, 22931745, 1346808619, 115763387962, 66757797341, 1379889220413, 15170474235886

Offset: 1

Robert G. Wilson v, Jul 07 2004

First differs from A050283 at n = 10. In particular, a(10) > a(11) while A050283(10) = A050283(11) = a(11). - Dmitry Petukhov, Jan 25 2020

a(14) > 22*10^12. - Dmitry Petukhov, Jan 25 2020

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

Cf. A050279, A035117, A096756, A096757, A050283, A096759, A096760, A096761, A096762, A096763 (similar for digits 0 through 9).

a(10)-a(13) from Dmitry Petukhov, Jan 25 2020

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

A096755 Index of first occurrence of exactly n consecutive '1's in a row 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

Robert G. Wilson v, Jul 07 2004

Presently identical to A035117.

It would be interesting to know the source for a(10) ~ 4*10^9, since the angio.net web site only searches up to 200M digits of Pi and even on subidiom.com only 2e9 digits of Pi are available. - M. F. Hasler, Apr 13 2019

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]
Peter Trüb, 22.4 trillion digits of pi

Cf. A035117, A096756, A096757, A096758, A096759, A096760, A096761, A096762, A096763, A096764.

a(11) from Giovanni Resta, Sep 30 2019
a(12) from Yasumasa Kanada, 2002 and a(13) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019
a(14) from Dmitry Petukhov, Sep 19 2022

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

A050282 a(n) is the starting position of the first occurrence of a string of at least n 3's in the decimal expansion of Pi.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Differs from A096757 which lists occurrences of strings of exactly n '3's. - M. F. Hasler, Mar 17 2017

a(14) > 50*10^12. - Dmitry Petukhov, Oct 28 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. A035117 (n '1's), 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).

a(9)-a(10) from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
a(11) from Giovanni Resta, Oct 02 2019
a(12) from Dmitry Petukhov, Jan 26 2020
a(13) from Dmitry Petukhov, Oct 28 2021

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

Original entry on oeis.org

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

A050287 Starting position of the first occurrence of a string of at least n '8's in the decimal expansion of Pi.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein

Differs from A096762 from a(3) = a(4) = A096762(4) < A096762(3) on. - M. F. Hasler, Mar 19 2017

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

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
a(11)-a(13) added by Dmitry Petukhov, Dec 30 2019
a(14) from Dmitry Petukhov, Sep 20 2022

A053753 Positions of 9's in the decimal expansion of Pi.

Original entry on oeis.org

6, 13, 15, 31, 39, 43, 45, 46, 56, 59, 63, 80, 81, 101, 123, 130, 145, 170, 181, 188, 191, 194, 200, 209, 215, 248, 250, 260, 285, 295, 329, 332, 337, 342, 354, 357, 389, 392, 400, 415, 417, 419, 423, 434, 441, 460, 461, 466, 483, 488, 497, 499, 502, 528, 530, 534, 543, 550

Offset: 1

Simon Plouffe, Feb 20 2000

Here "position" is 1, 2, 3, ... for the digits 3, 1, 4, ..., in contrast to A037007(n) = a(n)-1 and other sequences (A048940, A096763, ...) using "offset 0" for the position. - M. F. Hasler, Mar 20 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi
R. K. Hoeflin, Ultra Test. [Fixed broken link, replaced deleted page by web.archive backup, but the reference seems irrelevant. - _M. F. Hasler_, Mar 20 2017]

Cf. A037007, A048940, A096763, and the link to the OEIS index.

Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 9]] (* Vincenzo Librandi, Oct 07 2013 *)

default(realprecision,1999);for(i=2,#T=Vec(Str(Pi)),T[i-1]=="9"&&print1(i-2",")) \\ Exclude last digit from search. - M. F. Hasler, Mar 20 2017

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

cp's OEIS Frontend

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

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A096758 Index of first occurrence of exactly n consecutive fours in a row in the decimal expansion of Pi.

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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).

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A096755 Index of first occurrence of exactly n consecutive '1's in a row in the decimal expansion of Pi.

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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,...).

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A050282 a(n) is the starting position of the first occurrence of a string of at least n 3's in the decimal expansion of Pi.

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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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A050287 Starting position of the first occurrence of a string of at least n '8's in the decimal expansion of Pi.

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A053753 Positions of 9's in the decimal expansion of Pi.

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