A048940 -id:A048940 - cp's OEIS frontend

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

A081876 a(n) is the starting position of the second occurrence of a string of the initial n decimal digits of Pi in the decimal expansion of Pi.

Original entry on oeis.org

9, 137, 2120, 3496, 88008, 176451, 25198140, 50366472, 1660042751, 7902183159, 260816757309, 1142905318634, 17475119650043, 43420162171515

Offset: 1

Harry J. Smith, Apr 12 2003

The digits 3 1 4 1 5 ... are labeled 0, 1, 2, 3, 4, ...
a(9) > 10^8. - Robert G. Wilson v, May 09 2003
a(15) > 50*10^12. - Dmitry Petukhov, Oct 27 2021

Alfred S. Posamentier & Ingmar Lehmann, A Biography of the World's Most Mysterious Number, Prometheus Books, Amherst, NY 2004, page 134.

Dave Andersen, The Pi-Search Page
Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information.
Timothy Mullican, 50 trillion digits of pi
Harry J. Smith, Computing Pi [Broken link]
Harry J. Smith, Computing Pi
Peter Trüb, 22.4 trillion digits of pi
Pi search engine

Cf. A000796, A048940, A166028, A385936, A385937.

a(9) from Felix Fröhlich, Oct 04 2016
a(10)-a(11) from Andreas Stiller, Apr 08 2019
a(12) from Robert G. Wilson v, Oct 21 2004
a(13) from Dmitry Petukhov, Jan 27 2020
a(14) from Dmitry Petukhov, Oct 27 2021

A243955 a(n) is the starting position of the first occurrence of a string of n 9's in the decimal expansion of 2*Pi.

Original entry on oeis.org

12, 79, 761, 761, 761, 761, 761, 36356642, 1068822186

Offset: 1

Jose Luis Garcia del Castillo, Jun 19 2014

This sequence denotes the position of the Feynman point in the digits of tau = 2*Pi, from a(3) to a(7). Note how compared with A048940, tau has a Feynman point one digit longer, and one position lower, because at this point, the decimal expansion of Pi is 499999987, giving 99999997 when multiplied by 2.

			The position of one consecutive 9 in 2*Pi=6.283185307179 is on the 12th decimal digit.

Eric Weisstein's World of Mathematics, Feynman Point

Cf. A048940 (similar, with Pi), A019692 (decimal expansion of 2*Pi).

a(9) from Chai Wah Wu, Sep 30 2019

A256160 Index of the n-th 9 in the first occurrence of a string of exactly n 9's in the decimal expansion of Pi.

Original entry on oeis.org

6, 46, 2952, 17992, 19451, 768, 1722783, 36356650, 564665215, 20148132320, 27014073315, 897831316568, 10542036048463, 5758910552723

Offset: 1

Kival Ngaokrajang, Mar 16 2015

A048940 is 5, 44, 762, 762, 762, 762, 1722776, 36356642, 564665206, ... - Alonso del Arte, Mar 25 2015

Dave Andersen, The Pi-Search Page.

Cf. A048940, A096763, A256109.

{ default(realprecision, 20080); for (m = 1, 10, x = Pi; r = 0; for (n = 1, 20000, d = floor(x); x = (x-d)*10; if(d <> 9, if (r <> m, r = 0, print1(n-1, ", "); r = 0; break), r = r + 1)))}

a(n) = A096763(n) + n. - Danny Rorabaugh, Mar 31 2015

a(10)-a(14) added using A096763 by Jinyuan Wang, Mar 30 2020

A245331 Number of truncated Pi decimal digits that yield record approximations to Pi when the concatenation of the first half of the digits is divided by the second half.

Original entry on oeis.org

2, 23, 87, 157, 1523, 3445551, 26620870, 30512347, 72713283, 344661698, 1129330411, 3886591581, 5085084202, 11916345303, 15510679381

Offset: 1

Eric Angelini and Hans Havermann, Jul 18 2014

For odd terms, the number of digits in the first "half" is one more than in the second half. Even terms imply the second half begins with 1; odd terms, with 9.
The second-half numbers:
1
97932384626
99375105820974944592..
99862803482534211706..
99999983729780499510..
99999993176688420006..
10000000420467135547..
99999998414267344764..
99999999542282360035..
10000000012202360559..
99999999941927584272..
99999999948261395946..
10000000002413899137..
99999999975954453917..
99999999988383727123..

			a(1) is 2 because 3/1 (1+1 digits) provides the first approximation to Pi. a(2) is 23 because 314159265358/97932384626 (12+11 digits) provides the next better approximation.

Cf. A000796, A048940, A193940.

a(12)-a(15) from Hans Havermann, Jul 19 2014

A277535 Decimal expansion of Pi(10^761) - floor(Pi(10^761)).

Original entry on oeis.org

9, 9, 9, 9, 9, 9, 8, 3, 7, 2, 9, 7, 8, 0, 4, 9, 9, 5, 1, 0, 5, 9, 7, 3, 1, 7, 3, 2, 8, 1, 6, 0, 9, 6, 3, 1, 8, 5, 9, 5, 0, 2, 4, 4, 5, 9, 4, 5, 5, 3, 4, 6, 9, 0, 8, 3, 0, 2, 6, 4, 2, 5, 2, 2, 3, 0, 8, 2, 5, 3, 3, 4, 4, 6, 8, 5, 0, 3, 5, 2, 6, 1, 9, 3, 1, 1, 8, 8, 1, 7, 1, 0, 1, 0, 0, 0, 3, 1, 3, 7, 8, 3, 8, 7, 5

Offset: 0

Bobby Jacobs, Oct 19 2016

An approximation to 1.

There are 6 consecutive 9's starting at the 762nd decimal place of Pi. This sequence of six nines is also called "Feynman point" after physicist Richard Feynman.

			0.9999998372978049951059731732816096318595024459455346908302642522308253...
a(0) = A000796(763) = 9.
a(1) = A000796(764) = 9.
a(2) = A000796(765) = 9.
a(3) = A000796(766) = 9.
a(4) = A000796(767) = 9.
a(5) = A000796(768) = 9.

Eric Weisstein's World of Mathematics, Feynman Point
Wikipedia, Six nines in pi
Index entries for transcendental numbers

Cf. A000796, A048940, A243955.

With[{n = 762}, Drop[First@ RealDigits[N[Pi, n + 105]], n]] (* Michael De Vlieger, Oct 21 2016 *)
RealDigits[Pi, 10, 105, -762][[1]] (* Robert G. Wilson v, Oct 21 2016 *)

frac(Pi*10^761) \\ Charles R Greathouse IV, Oct 01 2022

a(n) = A000796(n+763).

cp's OEIS Frontend

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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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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A081876 a(n) is the starting position of the second occurrence of a string of the initial n decimal digits of Pi in the decimal expansion of Pi.

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A243955 a(n) is the starting position of the first occurrence of a string of n 9's in the decimal expansion of 2*Pi.

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A256160 Index of the n-th 9 in the first occurrence of a string of exactly n 9's in the decimal expansion of Pi.

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A245331 Number of truncated Pi decimal digits that yield record approximations to Pi when the concatenation of the first half of the digits is divided by the second half.

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A277535 Decimal expansion of Pi(10^761) - floor(Pi(10^761)).