A049522 -id:A049522 - cp's OEIS frontend

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

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

A049523 Smallest starting index for a run of exactly n consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

1, 25, 154, 1590, 17535, 763, 710101, 22931746, 24658602, 386980413, 15647738229, 368299898267, 2164164669333, 5758910552710, 46970519777308

Offset: 1

Harvey P. Dale

Digits 3,1,4,1,5,... are indexed 1,2,3,4,5,...
a(11) > 2*10^9. - M. F. Hasler, Mar 22 2017
a(12) > 99*10^9. - Giovanni Resta, Oct 02 2019
a(15) > 12*10^12. - Dmitry Petukhov, Dec 30 2019
a(16) > 50*10^12, a(17) = 28642224609576. - Dmitry Petukhov, Oct 30 2021

			See A084144 for an example.

Dave Andersen, Pi-Search Page
Yasumasa Kanada, Statistical Distribution Information, Home page, Computer Centre, The University of Tokyo.
Timothy Mullican, 50 trillion digits of pi
PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]
James Taylor, Irrational Number Search Engine
Peter Trüb, 22.4 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits.
Index entries for sequences related to the number Pi

Cf. A049514, A049515, A049516, A049517, A049518, A049519, A049520, A049521, A049522.

Cf. A084144 (digit with this starting index).

More terms from Rick L. Shepherd, May 15 2003
Edited and a(10) (via Taylor's search engine) added by M. F. Hasler, Mar 21 2017
a(11) from Giovanni Resta, Oct 02 2019
a(12)-a(14) added by Dmitry Petukhov, Dec 30 2019
a(15) from Dmitry Petukhov, Oct 30 2021

A049514 Starting index of a string of 2 or more consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

25, 35, 45, 60, 80, 95, 118, 126, 131, 136, 154, 155, 175, 178, 179, 183, 186, 202, 205, 212, 216, 218, 231, 258, 277, 283, 308, 310, 316, 318, 323, 361, 363, 365, 373, 378, 396, 402, 428, 438, 446, 454, 460, 473, 485, 495, 504, 508, 512, 517, 536, 560, 593

Offset: 1

Harvey P. Dale

Digits 3,1,4,... are indexed 1,2,3,...
A095916(a(n)) = 0. - Reinhard Zumkeller, Mar 12 2015
See A049518 for the "exactly 2" variant, which differs from a(11) on. - M. F. Hasler, Oct 18 2019

			From _M. F. Hasler_, Oct 18 2019: (Start)
The integer part of Pi*10^25 ends in 33, i.e., at position 25 starts the (first) string of two repeated digits 3, therefore a(1) = 25.
At position 154 starts a string of three '1's, so this sequence lists both, 154 and 155, but sequence A049518 lists none of these. (End)

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

Cf. A049515, A049516, A049517, A049518, A049519, A049520, A049521, A049522, A049523.

Cf. A000796, A095916.

a049514 n = a049514_list !! (n-1)
a049514_list = filter ((== 0) . a095916) [1..]
-- Reinhard Zumkeller, Mar 12 2015

ConsecutiveOccurrences1[alist_, n_] := Flatten @ Position[ Apply[ SameQ, Partition[ alist, n, 1], {1}], True]; ConsecutiveOccurrences1[ First[ RealDigits[Pi, 10, 601]], 2]
Flatten[Position[Partition[RealDigits[Pi,10,1000][[1]],2,1],?(#[[1]] == #[[2]]&),{1},Heads->False]] (* _Harvey P. Dale, Dec 21 2014 *)
SequencePosition[RealDigits[Pi,10,1000][[1]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 30 2019 *)

A049514_upto(N=999)={default(realprecision,N); my(p=digits(Pi\10^-N)); select(i->p[i]==p[i+1], [9..N-1])} \\ M. F. Hasler, Oct 18 2019

Edited by Robert G. Wilson v, May 09 2003

A049515 Starting index of a string of 3 or more consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

154, 178, 602, 763, 764, 765, 766, 856, 984, 1233, 1451, 1590, 1591, 1599, 1699, 1736, 1890, 2279, 2360, 2377, 2441, 2675, 2708, 2929, 2950, 3152, 3435, 3477, 3504, 3810, 3867, 3993, 4001, 4176, 4256, 4436, 4509, 4576, 4752, 4753, 4794, 4903, 4904, 4924

Offset: 1

Harvey P. Dale

Digits 3,1,4,... are indexed 1,2,3,...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A049514, A049516, A049517, A049518, A049519, A049520, A049521, A049522, A049523.

ConsecutiveOccurrences1[alist_, n_] := Flatten @ Position[ Apply[ SameQ, Partition[ alist, n, 1], {1}], True]; ConsecutiveOccurrences1[ First[ RealDigits[Pi, 10, 4928]], 3]
SequencePosition[RealDigits[Pi,10,5000][[1]],{x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 19 2019 *)

Edited by Robert G. Wilson v, May 09 2003

A049517 Starting index of a string of 5 or more consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

763, 764, 17535, 19447, 24467, 28468, 32789, 39862, 48440, 56989, 65261, 89086, 102388, 120460, 140745, 141900, 146044, 161863, 162249, 193035, 193036, 205035, 205194, 211059, 213246, 215288, 220569, 222300, 222301, 244454, 244455, 250130

Offset: 1

Harvey P. Dale

Digits 3,1,4,... are indexed 1,2,3,...

Cf. A049514, A049515, A049516, A049518, A049519, A049520, A049521, A049522, A049523.

ConsecutiveOccurrences1[alist_, n_] := Flatten @ Position[ Apply[ SameQ, Partition[ alist, n, 1], {1}], True]; ConsecutiveOccurrences1[ First[ RealDigits[Pi, 10, 252499]], 5]

Edited by Robert G. Wilson v, May 09 2003

A049516 Starting index of a string of 4 or more consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

763, 764, 765, 1590, 4752, 4903, 5242, 5323, 5864, 7965, 12487, 12701, 13391, 16733, 17535, 17536, 17989, 19438, 19447, 19448, 21881, 22754, 24467, 24468, 28468, 28469, 29505, 29869, 30797, 31901, 32428, 32479, 32789, 32790, 33108

Offset: 1

Harvey P. Dale

Digits 3,1,4,... are indexed 1,2,3,...

Cf. A049514, A049515, A049517, A049518, A049519, A049520, A049521, A049522, A049523.

ConsecutiveOccurrences1[alist_, n_] := Flatten @ Position[ Apply[ SameQ, Partition[ alist, n, 1], {1}], True]; ConsecutiveOccurrences1[ First[ RealDigits[Pi, 10, 33172]], 4]
SequencePosition[RealDigits[Pi,10,34000][[1]],{x_,x_,x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 16 2019 *)

Edited by Robert G. Wilson v, May 09 2003

A049534 Starting index of a string of 6 or more consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

763, 193035, 222300, 244454, 252500, 253210, 255946, 399580, 419998, 452072, 710101, 828500, 963025, 1006928, 1129020, 1264271, 1637081, 1691164, 1699928, 1722777, 1795774, 1985814, 2309219, 2328784, 2376568, 2418534, 2523357

Offset: 1

Harvey P. Dale

Digits 3,1,4,... are indexed 1,2,3,... (in contrast to, e.g., A083600 - A083645).
The successive strings are 6 nines, 6 nines, 6 eights, 6 fives, 6 sixes, 6 fives, 6 ones, 6 sevens, 6 fives, 6 sevens, 7 threes, 6 fours, 6 twos, 6 sevens, 6 threes, 6 fours, 6 twos, 6 fours, 6 zeros, 7 nines, 6 twos, 6 nines, 6 sevens, 6 zeros, 6 sevens, 6 eights, 6 twos, 6 zeros, 6 ones, 6 nines, 6 eights, 6 nines, 6 eights, 7 threes, 6 ones, 6 fours, 6 fours, 7 sevens, 7 nines, 6 twos, 7 fives, 6 nines, 6 fours, 6 eights, 7 sevens, 7 zeros, 6 sixes, 6 threes, 6 sixes, 7 nines, 6 sevens, 6 threes, 7 ones, 7 eights, ..., . - Robert G. Wilson v, Aug 28 2006
If there are more than 6 equal digits starting at a(n), then a(n)+1 etc. is not listed, in contrast to, e.g., A083600 - A083645, and most other sequences of this type. Therefore the sequence data yields only candidates for longer runs, but they cannot be deduced from the data as this sequence can be deduced from consecutive numbers in A049517, cf. formula. - M. F. Hasler, Mar 21 2017

David G. Andersen, The Pi-Search Page.
Index entries for sequences related to the number Pi

Cf. A049514, A049515, A049516, A049517: starting positions of 2, 3, 4, 5 consecutive equal digits; A049518, A049519, A049520, A049521: exactly 2, 3, 4, 5 consecutive equal digits, A049522, A049523: first occurrence of (at least / exactly) n consecutive equal digits.
Cf. A083600, A083604, A083609, A083613, A083618, A083623, A083628, A083634, A083640, and A083645: starting positions of 6 consecutive '0's, ..., '9's.
Cf. A049517: starting position of 5 or more consecutive equal digits.

p = RealDigits[Pi, 10, 2645268][[1]]; Select[ Range@2645263, p[[ # ]] == p[[ # + 1]] == p[[ # + 2]] == p[[ # + 3]] == p[[ # + 4]] == p[[ # + 5]] &]; (* Robert G. Wilson v, Aug 28 2006 *)

Union of A083600, A083604, A083609, A083613, A083618, A083623, A083628, A083634, A083640, and A083645, plus one (because of indexing convention), and consecutive numbers removed in each of the sets. Also, { A049517(n) | A049517(n+1) = A049517(n)+1, but not A049517(n-1) = A049517(n)-1 } = { n+1 | (floor(Pi*10^n) mod 10^6) mod 111111 = 0, but not for n-1 }, where mod is the binary "remainder" operator. - M. F. Hasler, Mar 21 2017

Entry revised by N. J. A. Sloane, Aug 26 2006

More terms from Robert G. Wilson v, Aug 28 2006

A083630 Starting positions of strings of eight '6's in the decimal expansion of Pi.

Original entry on oeis.org

45681781, 45681782, 55616210, 129423072, 160301327, 320800854, 354039866, 386980412, 386980413, 386980414, 424213905, 424213906, 661180039, 823960247, 864397328, 923385691, 1064174618, 1129615231, 1183548825, 1208233597, 1261092402, 1311369257

Offset: 1

Rick L. Shepherd, May 04 2003

Digits 3,1,4,... are indexed 0,1,2,... (as in the two search engines and most other related sequences, but not, e.g., in A049522).

Dave Andersen, Pi-Search Page
James Taylor, Irrational Number Search Engine
Eric Weisstein's World of Mathematics, Pi Digits.
Index entries for sequences related to the number Pi

Cf. A083629 (seven '6's), A083636 (eight '7's).

Cf. A049522, A049523.

a(4) and a(5) from Harvey P. Dale, May 05 2011

More terms, via Taylor's search engine, from M. F. Hasler, Mar 21 2017

A083636 Starting positions of strings of eight 7's in the decimal expansion of Pi.

Original entry on oeis.org

24658601, 24658602, 82144203, 287248607, 1096547723, 1136093224, 1136093225, 1199887625, 1253506894, 1284344221, 1284344222, 1343455078, 1347842020, 1387894932, 1387894933, 1439845247, 1577148578

Offset: 1

Rick L. Shepherd, May 05 2003

From M. F. Hasler, Mar 21 2017: (Start)
Positions 0,1,2,... correspond to digits 3,1,4,... as in the search engines and most related sequences, while some sequences, e.g., A049522, use indexing starting at 1.
Consecutive numbers indicate that there are more than eight '7's in a row, e.g., at positions 24658601, 1136093224, 1387894932,... start runs of nine '7's. The first of these is also A049522(9)-1 = A049523(9)-1. (End)

Dave Andersen, Pi-Search Page
James Taylor, Irrational Number Search Engine
Eric Weisstein's World of Mathematics, Pi Digits.
Index entries for sequences related to the number Pi

Cf. A083635 (seven '7's), A083630 (eight '6's).

Terms beyond a(3) from M. F. Hasler, Mar 21 2017

A084073 Starting index of a record-breaking string of consecutive equal digits in the decimal expansion of Pi.

Original entry on oeis.org

1, 25, 154, 763, 710101, 22931746, 24658602, 386980413

Offset: 1

Robert G. Wilson v, May 09 2003

Digits 3,1,4,... are indexed 1,2,3,...,.

			The repeated digits are 3, 3, 1, 6, 3, 4, 7, ..., i.e., at position a(1) = 1, there is one '3', starting at position a(2) = 25, there are two '3's, from position a(3) = 154 on, there are three '1's, etc.

Index entries for sequences related to the number Pi

Positions of records in A082586.

Records in A049522. See also A049523, A049534.

Edited and a(10) from A049522 added by M. F. Hasler, Mar 21 2017

cp's OEIS Frontend

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

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A049523 Smallest starting index for a run of exactly n consecutive equal digits in decimal expansion of Pi.

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A049514 Starting index of a string of 2 or more consecutive equal digits in decimal expansion of Pi.

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A049515 Starting index of a string of 3 or more consecutive equal digits in decimal expansion of Pi.

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A049517 Starting index of a string of 5 or more consecutive equal digits in decimal expansion of Pi.

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A049516 Starting index of a string of 4 or more consecutive equal digits in decimal expansion of Pi.

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A049534 Starting index of a string of 6 or more consecutive equal digits in decimal expansion of Pi.

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A083630 Starting positions of strings of eight '6's in the decimal expansion of Pi.

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