cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Harvey P. Dale

Keywords

Comments

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

Examples

			See A084144 for an example.

Links

Crossrefs

Cf. A049514, A049515, A049516, A049517, A049518, A049519, A049520, A049521, A049522.
Cf. A084144 (digit with this starting index).

Extensions

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

Views

Author

Harvey P. Dale

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

Cf. A049515, A049516, A049517, A049518, A049519, A049520, A049521, A049522, A049523.
Cf. A000796, A095916.

Programs

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

Extensions

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

Views

Author

Harvey P. Dale

Keywords

Comments

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

Crossrefs

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

Programs

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

Extensions

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

Views

Author

Harvey P. Dale

Keywords

Comments

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Harvey P. Dale

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

Extensions

Entry revised by N. J. A. Sloane, Aug 26 2006
More terms from Robert G. Wilson v, Aug 28 2006
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.