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Search: a121280 -id:a121280
Displaying 1-10 of 14 results found. page 1 2
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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. +0
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
A037001 Positions of the digit '2' in the decimal expansion of Pi (where positions 0, 1, 2,... refer to the digits 3, 1, 4,...). +0
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
A048940 Starting position of the first occurrence of a string of at least n '9's in the decimal expansion of Pi. +0
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
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. +0
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
A050286 Starting position of the first occurrence of a string of at least n '7's in the decimal expansion of Pi. +0
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. +0
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
A053746 Positions of '2's in the decimal expansion of Pi, where positions 1, 2, 3, ... correspond to digits 3, 1, 4, ... +0
7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
See A037001 for the variant where digits 3, 1, 4, ... correspond to positions 0, 1, 2, ... - M. F. Hasler, Jul 28 2024
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to the number Pi
FORMULA
a(n) = A037001(n) + 1. - Georg Fischer, May 31 2021
EXAMPLE
Pi = 3.1415926... where the first '2' occurs as the 7th digit.
MATHEMATICA
Flatten[Position[RealDigits[Pi, 10, 1000][[1]], 2]] (* Vincenzo Librandi, Oct 07 2013 *)
PROG
(PARI) A053746_upto(N=999)={localprec(N+20); select(d->d==2, digits(Pi\10^-N), 1)} \\ M. F. Hasler, Jul 28 2024
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).
KEYWORD
nonn,base
AUTHOR
Simon Plouffe, Feb 20 2000
EXTENSIONS
Changed offset from 0 to 1 by Vincenzo Librandi, Oct 07 2013
STATUS
approved
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). +0
15
2, 149, 1925, 13808, 49703, 2458886, 9470345, 186557267, 523551503, 191278379840, 4368196101672 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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
Table of n, a(n) for n=1..11.
D. G. Andersen, The Pi-Search Page
SubIdiom.com, Irrational numbers search engine: π = 3.14159.... (Search within 2*10^9 digits, since at least 2009, maybe 2002.)
Peter Trüb, 22.4 trillion digits of pi
FORMULA
a(n) = A121280(n) + 1. - M. F. Hasler, Apr 13 2019
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}]
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
Joseph L. Pe, Apr 01 2002
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
STATUS
approved
A096761 Position of first occurrence of exactly n consecutive sevens in a row in the decimal expansion of Pi. +0
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. +0
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
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Last modified August 30 11:38 EDT 2024. Contains 375543 sequences. (Running on oeis4.)