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.

Previous Showing 11-20 of 20 results.

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

Views

Author

Robert G. Wilson v, Jul 07 2004

Keywords

Comments

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

Links

Crossrefs

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

Extensions

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

Views

Author

Joseph L. Pe, Apr 01 2002

Keywords

Comments

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

References

  • Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, p. 32.

Links

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

a(n) = A121280(n) + 1. - M. F. Hasler, Apr 13 2019

Extensions

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

Views

Author

Robert G. Wilson v, Jul 07 2004

Keywords

Comments

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

Links

Crossrefs

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

Extensions

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

Views

Author

Alexander R. Povolotsky, Nov 03 2007

Keywords

Comments

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

Links

Crossrefs

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.

Formula

a(n) = A068987(n) - 1.

Extensions

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

A050281 a(n) is the starting position of the first occurrence of a string of n 2's in the decimal expansion of Pi.

Original entry on oeis.org

6, 135, 1735, 4902, 65260, 963024, 82599811, 175820910, 1270311937, 20717271655, 225023890967, 1479132847647, 5547233660249
Offset: 1

Views

Author

Eric W. Weisstein

Keywords

Links

Crossrefs

Cf. A035117.

Extensions

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002
a(10) from Giovanni Resta, Oct 02 2019
a(11)-a(13) from Dmitry Petukhov, Jan 14 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

Views

Author

Eric W. Weisstein

Keywords

Comments

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

Links

Crossrefs

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).

Extensions

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

Views

Author

Eric W. Weisstein

Keywords

Comments

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

Links

Crossrefs

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)).

Formula

a(n) = min { A096761(k); k >= n }. - M. F. Hasler, Mar 19 2017

Extensions

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

Views

Author

Eric W. Weisstein

Keywords

Comments

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

Links

Crossrefs

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.

Extensions

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

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

Views

Author

Simon Plouffe, Feb 20 2000

Keywords

Comments

See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024

Examples

			Pi = 3.1415926... where the first '2' occurs as the 7th digit.

Links

Crossrefs

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).

Programs

  • Mathematica
    Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)
  • PARI
    A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024

Formula

a(n) = A037001(n) + 1. - Georg Fischer, May 31 2021

Extensions

Changed offset from 0 to 1 by Vincenzo Librandi, Oct 07 2013

A178709 Position of start of first appearance of n consecutive 1's in the binary expansion of Pi.

Original entry on oeis.org

3, 11, 11, 11, 11, 11, 451, 645, 645, 645, 5212, 18123, 18123, 58276, 58276, 80697, 80697, 80697, 1146746, 1962901, 3296306, 9772065, 9772065, 9772065, 47536571, 169338693, 169338693, 207861698, 207861698, 207861698
Offset: 1

Views

Author

Will Nicholes, Jun 06 2010

Keywords

Comments

Out of the first 2^28 binary digits, 134220460 are "0" and 134214996 are "1". - Robert G. Wilson v, Jun 09 2010
This sequence ignores bits in the integer part of the binary expansion of Pi.

Examples

			6 consecutive 1's are first found beginning at the 11th position in Pi's binary expansion, so the sixth term in this sequence is 11.

Crossrefs

Cf. A004601, A035117, A178708.

Programs

  • Mathematica
    pib = ToString@ FromDigits[ RealDigits[Pi - 3, 2, 2^28][[1]]]; f[n_] := 2 + StringPosition[ pib, ToString[(10^n - 1)/9], 1][[1, 1]]; Array[f, 30] (* Robert G. Wilson v, Jun 09 2010 *)

Extensions

a(14)-a(30) from Robert G. Wilson v, Jun 09 2010
Previous Showing 11-20 of 20 results.

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

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