A049523 -id:A049523 - cp's OEIS frontend

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

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

A049522 Smallest starting index for a run of at least n consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

1, 25, 154, 763, 763, 763, 710101, 22931746, 24658602, 386980413, 15647738229, 368299898267, 2164164669333, 5758910552710, 28642224609577, 28642224609577, 28642224609577

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 21 2017
a(12) > 99*10^9. - Giovanni Resta, Oct 02 2019
a(15) > 12*10^12. - Dmitry Petukhov, Dec 30 2019
a(18) > 50*10^12. - Dmitry Petukhov, Oct 30 2021

Pi search engine (searches initial 2 x 10^9 digits of Pi)
Dave Andersen, Pi-Search Page (searches initial 2 x 10^8 digits of Pi)
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]
Peter Trüb, 22.4 trillion digits of pi
Eric Weisstein's World of Mathematics, Pi Digits.

Cf. A049514-A049523.
Compare A049523: the first run of six 9's occurs (with starting index 763) before any runs of exactly four or exactly five equal digits.
Cf. A084145 (digit with this starting index).

More terms from Rick L. Shepherd, May 22 2003
a(10) from Felix Fröhlich, Oct 06 2016
a(11) from Giovanni Resta, Oct 02 2019
a(12)-a(14) added by Dmitry Petukhov, Dec 30 2019
a(15)-a(17) from Dmitry Petukhov, Oct 30 2021
a(15)-a(17) had off-by-1 error, corrected by Mike Keith, Feb 03 2025

A049518 Starting index of a string of exactly 2 consecutive equal digits in the decimal expansion of Pi.

Original entry on oeis.org

25, 35, 45, 60, 80, 95, 118, 126, 131, 136, 175, 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, 622

Offset: 0

Harvey P. Dale

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

See A049514 for the variant "at least 2", 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*10^22 ends in 33, i.e., at position 22 starts the (first) string of two repeated digits 3, therefore a(1) = 22.
At position 154 starts a string of three '1's, so sequence A049514 lists both, 154 and 155, but this sequence lists none of these. (End)

Cf. A049514-A049523.

Flatten[Position[Partition[RealDigits[Pi,10,1000][[1]],4,1],?(#[[1]] != #[[2]] && #[[2]]==#[[3]]&&#[[3]]!=#[[4]]&),1,Heads->False]]+1 (* _Harvey P. Dale, Jul 08 2017 *)

A049518_upto(N=999)={default(realprecision, N); my(p=digits(Pi\10^-N)); select(i->p[i]==p[i+1] && p[i]!=p[i-1] && p[i]!=p[i+2], [9..N-2])} \\ M. F. Hasler, Oct 18 2019

A049519 Starting index of a string of exactly 3 consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

154, 178, 602, 856, 984, 1233, 1451, 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, 4794, 4924, 4929, 4986, 5291, 5356, 5404, 5451, 5676, 5872, 6071

Offset: 0

Harvey P. Dale

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

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A049514-A049523.

e3cdQ[{a_,b_,c_,d_,e_}]:=a!=b&&d!=e&&b==c==d; Flatten[Position[ Partition[ RealDigits[Pi,10,7000][[1]],5,1],?(e3cdQ)]]+1 (* _Harvey P. Dale, Nov 09 2016 *)

A049521 Starting index of a string of exactly 5 consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

17535, 19447, 24467, 28468, 32789, 39862, 48440, 56989, 65261, 89086, 102388, 120460, 140745, 141900, 146044, 161863, 162249, 205035, 205194, 211059, 213246, 215288, 220569, 250130, 266895, 283694, 309380, 322348, 327075, 335793, 363996

Offset: 0

Harvey P. Dale

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

Cf. A049514-A049523.

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

A049520 Starting index of a string of exactly 4 consecutive equal digits in decimal expansion of Pi.

Original entry on oeis.org

1590, 4752, 4903, 5242, 5323, 5864, 7965, 12487, 12701, 13391, 16733, 17989, 19438, 21881, 22754, 29505, 29869, 30797, 31901, 32428, 32479, 33108, 33173, 35458, 37231, 37323, 40793, 42096, 43155, 43406, 43524, 46513, 48666, 49056

Offset: 0

Harvey P. Dale

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

Cf. A049514-A049523.

Flatten[Position[Partition[RealDigits[Pi,10,50000][[1]],6,1],?(Union[ Differences[Most[Rest[#]]]]=={0}&&First[Differences[#]]!=0 && Last[ Differences[#]]!=0&),{1},Heads->False]]+1 (* _Harvey P. Dale, Jun 02 2014 *)

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

cp's OEIS Frontend

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

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A049522 Smallest starting index for a run of at least n consecutive equal digits in decimal expansion of Pi.

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A049518 Starting index of a string of exactly 2 consecutive equal digits in the decimal expansion of Pi.

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A049519 Starting index of a string of exactly 3 consecutive equal digits in decimal expansion of Pi.

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A049521 Starting index of a string of exactly 5 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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A049520 Starting index of a string of exactly 4 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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