A068987 -id:A068987

A037001

Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...).

+10
26

6, 16, 21, 28, 33, 53, 63, 73, 76, 83, 89, 93, 102, 112, 114, 135, 136, 140, 149, 160, 165, 173, 185, 186, 203, 221, 229, 241, 244, 260, 275, 280, 289, 292, 298, 302, 326, 329, 333, 335, 337, 354, 374, 380, 406, 423, 435, 456, 462, 477, 479, 484, 485, 500

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The first few primes in this sequence are 53, 73, 83, 89, 149, 173, 229, 241, 337, 479, 571, 613, 661, 757, 829, 877, 911, 977, 991, ... - M. F. Hasler, Jul 28 2024

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pi Digits.

OEIS index to sequences related to Pi.

FORMULA

a(n) ~ 10*n if Pi is normal, as generally assumed. - M. F. Hasler, Jul 28 2024

MATHEMATICA

Flatten @ Position[ RealDigits[Pi - 3, 10, 500][[1]], 2] (* Robert G. Wilson v, Mar 07 2011 *)

PROG

(PARI) A037001_upto(N=999, d=2)={localprec(N+20); [i-1|i<-[1..#N=digits(Pi\10^-N)], N[i]==d]} \\ M. F. Hasler, Jul 28 2024

CROSSREFS

Cf. A000796 (decimal expansion (or digits) of Pi).

Cf. A053746 (= a(n) + 1: the same with different offset).

Cf. A037000, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008 (similar for digits 1, ..., 9 and 0).

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.

KEYWORD

nonn,base

AUTHOR

Nicolau C. Saldanha (nicolau(AT)mat.puc-rio.br)

STATUS

approved

A096763

Position of the first occurrence of exactly n consecutive '9's in a row in the decimal expansion of Pi.

+10
24

5, 44, 2949, 17988, 19446, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 10542036048450, 5758910552709

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..14.

David G. Andersen, The Pi-Search Page.

Yasumasa Kanada, Statistical Distribution Information, Home Page, Computer Centre, The University of Tokyo.

PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]

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 n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

KEYWORD

nonn,base,more

AUTHOR

Robert G. Wilson v, Jul 07 2004

EXTENSIONS

a(10)-a(11) from Giovanni Resta, Sep 30 2019

a(12) from Yasumasa Kanada, 2002 and a(13)-a(14) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019

STATUS

approved

A035117

a(n) is the starting position of the first occurrence of a string of at least n 1's in the decimal expansion of Pi.

+10
21

1, 94, 153, 12700, 32788, 255945, 4657555, 159090113, 812432526, 3961184001, 15647738228, 1041032609981, 3907688331257, 68635742334547

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Presently identical to A096755, which is the first occurrences of exactly n 1's in the digits of Pi. Will differ as soon as there's some a(n) = a(n+1) and equivalently, A035117(n) > A035117(n+1). - M. F. Hasler, Mar 17 2017

LINKS

Table of n, a(n) for n=1..14.

David G. Andersen, The Pi-Search Page.

PI-world Site, The digits and Statistics for 12 trillion digits of PI [archived page]

Eric Weisstein's World of Mathematics, Pi Digits.

Yasumasa Kanada Laboratory Home Page, Computer Centre, The University of Tokyo, Statistical Distribution Information

CROSSREFS

Cf. A000796 (decimal expansion (or digits) of Pi).

Cf. A035117 (this), 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).

Cf. A121280 = A068987 - 1 (position of "123...n" in Pi's decimals).

Cf. A176341 (first occurrence of n in Pi's digits).

KEYWORD

nonn,base,more

AUTHOR

Leonardo Bitran (lbitran(AT)reuna.cl)

EXTENSIONS

More terms from Colin Martin (cbmartin(AT)tpg.com.au), Mar 03 2002

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

Edited, after re-establishing A096755, by M. F. Hasler, Mar 17 2017

a(11) from Giovanni Resta, Sep 30 2019

a(12) from Yasumasa Kanada Laboratory, 2002 and a(13) from Shigeru Kondo, 2011, added by Dmitry Petukhov, Dec 27 2019

a(14) from Dmitry Petukhov, Sep 19 2022

STATUS

approved

A050279

a(n) is the starting position of the first occurrence of a string of at least n '0's in the decimal expansion of Pi.

+10
21

32, 307, 601, 13390, 17534, 1699927, 3794572, 172330850, 2542542102, 8324296435, 371247087572, 1755524129973, 3186699229890, 6381820482331

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

At least up to a(10), also the starting position of the first occurrence of a string of exactly n '0's in the decimal expansion of Pi, cf. A096764. - M. F. Hasler, Mar 19 2017, edited Sep 03 2017

a(15) > 22*10^12. - Dmitry Petukhov, Jan 28 2020

REFERENCES

Shigeru Kondo, calculation of Pi to 12.8 * 10^9 digits, using the program PiFast of Xavier Gourdon

LINKS

Table of n, a(n) for n=1..14.

David G. Andersen, The Pi-Search Page.

Peter Trüb, 22.4 trillion digits of pi

Eric Weisstein's World of Mathematics, Pi Digits

CROSSREFS

See A096764 for another version.

Cf. A000796: Decimal expansion (or digits) of Pi.

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), A096764 (exactly n '0's).

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 concatenate(1,...,n): A121280 = A068987 - 1.

KEYWORD

nonn,base,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

More terms from Colin B. Martin (martinc(AT)ram.net.au), Nov 25 2001

Edited by N. J. A. Sloane at the suggestion of M. F. Hasler, Aug 24 2007

Edited by M. F. Hasler, Mar 19 2017

Definition modified by N. J. A. Sloane, Sep 03 2017

a(11)-a(14) added by Dmitry Petukhov, Jan 12 2020

STATUS

approved

A096761

Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi.

+10
21

13, 559, 4575, 1589, 162248, 399579, 3346228, 82144203, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Differs from A050286 from a(3) > a(4) on. - M. F. Hasler, Mar 18 2017

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

Table of n, a(n) for n=1..15.

David G. Andersen, The Pi-Search Page.

Timothy Mullican, 50 trillion digits of pi

Peter Trüb, 22.4 trillion digits of pi

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).

KEYWORD

nonn,base,more

AUTHOR

Robert G. Wilson v, Jul 07 2004

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

STATUS

approved

A096762

Position of first occurrence of exactly n consecutive '8's in a row in the decimal expansion of Pi.

+10
21

11, 34, 4985, 4751, 213245, 222299, 4722613, 239798471, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(8) > 2*10^8, a(9) = 46663520, a(10) = 3040319543.

Differs from A050287 from a(3) > A050287(3) = A050287(4) = a(4) on. - M. F. Hasler, Mar 19 2017

LINKS

Table of n, a(n) for n=1..14.

David G. Andersen, The Pi-Search Page.

Peter Trüb, 22.4 trillion digits of pi

CROSSREFS

Cf. A000796: Decimal expansion (or digits) of Pi.

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 times the same digit: A035117 (n '1's), A050281 (n '2's), A050282, A050283, A050284, A050286, A050287, A048940 (n '9's).

First occurrence of concatenate(1,...,n): A121280 = A068987 - 1.

KEYWORD

base,nonn,more

AUTHOR

Robert G. Wilson v, Jul 07 2004

EXTENSIONS

Edited by M. F. Hasler, Mar 19 2017

a(8) via SubIdiom.com/pi search engine from M. F. Hasler, Apr 13 2019

a(11)-a(13) added by Dmitry Petukhov, Dec 30 2019

a(14) from Dmitry Petukhov, Sep 20 2022

STATUS

approved

A048940

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

+10
20

5, 44, 762, 762, 762, 762, 1722776, 36356642, 564665206, 20148132310, 27014073304, 897831316556, 5758910552709, 5758910552709

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..14.

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

MATHEMATICA

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 *)

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 n: A176341; of concatenate(1,...,n): A121280 = A068987 - 1.

Cf. A049522, A084073.

KEYWORD

nonn,base,more

AUTHOR

Eric W. Weisstein

EXTENSIONS

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

STATUS

approved

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,...).

+10
15

1, 148, 1924, 13807, 49702, 2458885, 9470344, 186557266, 523551502, 191278379839, 4368196101671

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..11.

Dave G. Andersen, The Pi-Search Page

Subidiom.com, Irrational numbers search engine: π = 3.14159...

Peter Trüb, 22.4 trillion digits of pi

FORMULA

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

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.

KEYWORD

nonn,base,more

AUTHOR

Alexander R. Povolotsky, Nov 03 2007

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

STATUS

approved

A050286

Starting position of the first occurrence of a string of at least n '7's in the decimal expansion of Pi.

+10
13

13, 559, 1589, 1589, 162248, 399579, 3346228, 24658601, 24658601, 22869046249, 165431035708, 368299898266, 10541103245815, 14793486898235, 46970519777308

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

Table of n, a(n) for n=1..15.

Timothy Mullican, 50 trillion digits of pi

Peter Trüb, 22.4 trillion digits of pi

Eric Weisstein's World of Mathematics, Pi Digits.

FORMULA

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

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)).

KEYWORD

nonn,base,more

AUTHOR

Eric W. Weisstein

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

STATUS

approved

A050287

Starting position of the first occurrence of a string of at least n '8's in the decimal expansion of Pi.

+10
13

11, 34, 4751, 4751, 213245, 222299, 4722613, 46663520, 46663520, 3040319543, 159999448572, 1141385905180, 2164164669332, 91250566353705

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..14.

Peter Trüb, 22.4 trillion digits of pi

Eric Weisstein's World of Mathematics, Pi Digits.

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.

KEYWORD

nonn,base,more

AUTHOR

Eric W. Weisstein

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

STATUS

approved