A037000 -id:A037000 - cp's OEIS frontend

A125628 Version of sexagesimal expansion of 2*Pi given by the Persian mathematician Al-Kashi in the 15th Century.

6, 16, 59, 28, 1, 34, 51, 46, 14, 50

Offset: 1

Mohammad K. Azarian, Apr 08 2008

The last digit is rounded up from 49, 55 (cf. A091649). - Georg Fischer, Aug 04 2021

			2*Pi ~= 6; 16, 59, 28, 1, 34, 51, 46, 14, 50.

Mohammad K. Azarian, Al-Risala al-Muhitiyya: A Summary, Missouri Journal of Mathematical Sciences, Vol. 22, No. 2, 2010, pp. 64-85.
Mohammad K. Azarian, The Introduction of Al-Risala al-Muhitiyya: An English Translation, International Journal of Pure and Applied Mathematics, Vol. 57, No. 6, 2009 , pp. 903-914.
Mohammad K. Azarian, Al-Kashi's Fundamental Theorem, International Journal of Pure and Applied Mathematics, Vol. 14, No. 4, 2004, pp. 499-509. Mathematical Reviews, MR2005b:01021 (01A30), February 2005, p. 919. Zentralblatt MATH, Zbl 1059.01005.
Mohammad K. Azarian, Meftah al-hesab: A Summary, MJMS, Vol. 12, No. 2, Spring 2000, pp. 75-95. Mathematical Reviews, MR 1 764 526. Zentralblatt MATH, Zbl 1036.01002.
Mohammad K. Azarian, A Summary of Mathematical Works of Ghiyath ud-din Jamshid Kashani, Journal of Recreational Mathematics, Vol. 29(1), pp. 32-42, 1998.

Wikipedia, Computation of 2 Pi

Cf. A000796, A011545, A037000, A091649.

Cf. A159991. - Reinhard Zumkeller, May 02 2009

A134210 Positions of 10 after the decimal point in the decimal expansion of Pi.

Original entry on oeis.org

49, 163, 175, 206, 269, 442, 681, 780, 852, 854, 1011, 1219, 1223, 1270, 1318, 1487, 1816, 1892, 2162, 2238, 2514, 2534, 2563, 2721, 2749, 2780, 2810, 2874, 2880, 2955, 3170, 3201, 3208, 3241, 3254, 3405, 3457, 3480, 3486, 3494, 3845, 3848, 3939, 3964, 3966

Offset: 1

Artur Jasinski, Oct 14 2007

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000796 (Pi).

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777.

SequencePosition[RealDigits[Pi,10,10000][[1]],{1,0}][[All,1]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 20 2016 *)

More terms from Harvey P. Dale, Nov 20 2016

A256521 Table T(n, k) of positions p[i] where number n occurs after the decimal point in the decimal expansion of Pi, read by antidiagonals.

Original entry on oeis.org

32, 50, 1, 54, 3, 6, 65, 37, 16, 9, 71, 40, 21, 15, 2, 77, 49, 28, 17, 19, 4, 85, 68, 33, 24, 23, 8, 7, 97, 94, 53, 25, 36, 10, 20, 13, 106, 95, 63, 27, 57, 31, 22, 29, 11, 116, 103, 73, 43, 59, 48, 41, 39, 18, 5, 121, 110, 76, 46, 60, 51, 69, 47, 26, 12, 49, 128, 138, 83, 64, 70, 61, 72, 56, 34, 14, 163, 94

Offset: 0

Felix Fröhlich, Apr 01 2015

Table T(n, k) starts:
n =  0: 32,  50,  54,  65,  71,  77,  85,  97, 106, 116, ...
n =  1:  1,   3,  37,  40,  49,  68,  94,  95, 103, 110, ...
n =  2:  6,  16,  21,  28,  33,  53,  63,  73,  76,  83, ...
n =  3:  9,  15,  17,  24,  25,  27,  43,  46,  64,  86, ...
n =  4:  2,  19,  23,  36,  57,  59,  60,  70,  87,  92, ...
n =  5:  4,   8,  10,  31,  48,  51,  61,  90, 109, 130, ...
n =  6:  7,  20,  22,  41,  69,  72,  75,  82,  98, 108, ...
n =  7: 13,  29,  39,  47,  56,  66,  96,  99, 120, 139, ...
n =  8: 11,  18,  26,  34,  35,  52,  67,  74,  78,  81, ...
n =  9:  5,  12,  14,  30,  38,  42,  44,  45,  55,  58, ...
n = 10: 49, 163, 175, 206, 269, 442, 681, 780, 852, 854, ...
...

			T(6, 4) = 41, since the fourth occurrence of 6 in the decimal expansion of Pi is at position 41.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
D. G. Andersen, The Pi-Search Page

Cf. A000796 (Pi), A014777 (first column).
Cf. A037008, A037000, A037001, A037002, A037003 (0th to 4th row).
Cf. A037004, A037005, A036974, A037006, A037007 (5th to 9th row).

spi = StringDrop[ ToString[ N[ Pi, 1000]], 2]; t[n_, k_] := StringPosition[ spi, ToString[n], k][[-1, 1]]; Table[ t[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Robert G. Wilson v, Apr 07 2015 *)

More terms from Robert G. Wilson v, Apr 07 2015

A332084 Triangle read by rows: T(n,k) is the smallest m >= 0 such that floor(Pi*n^m) == k (mod n), -1 if one does not exist, k = 0..n-1.

Original entry on oeis.org

0, 1, 0, 0, 2, 4, 1, 3, 2, 0, 1, 7, 3, 0, 8, 1, 9, 14, 0, 10, 2, 1, 7, 10, 0, 8, 6, 2, 3, 1, 8, 0, 9, 6, 14, 5, 10, 1, 2, 0, 3, 20, 18, 11, 5, 32, 1, 6, 0, 2, 4, 7, 13, 11, 5, 5, 1, 8, 0, 13, 4, 2, 6, 9, 24, 12, 5, 1, 22, 0, 3, 17, 14, 18, 2, 6, 20, 10, 5, 1, 10, 0, 6, 9, 17, 14, 23, 7, 2, 21, 3

Offset: 1

Davis Smith, Aug 22 2020

Pi is normal in base n >= 2 if and only if in every row N, such that N is a power of n, -1 does not appear. Pi is absolutely normal if and only if -1 never appears.
Conjecture: Pi is absolutely normal, meaning that -1 will never appear.
This triangle is an instance of the more general f(n,k,r), where f(n,k,r) is the smallest m >= 0 such that floor(r*n^m) == k (mod n) (-1 if one does not exist) and r is irrational. The same conditions for normalcy apply.

			The triangle T(n,k) starts:
n\k   0   1   2   3   4   5   6   7   8   9  10  11  12 ...
1:    0
2:    1   0
3:    0   2   4
4:    1   3   2   0
5:    1   7   3   0   8
6:    1   9  14   0  10   2
7:    1   7  10   0   8   6   2
8:    3   1   8   0   9   6  14   5
9:   10   1   2   0   3  20  18  11   5
10:  32   1   6   0   2   4   7  13  11   5
11:   5   1  22   0  13   4   2   6   9  24  12
12:   5   1  10   0   3  17  14  18   2   6  20  10
13:   5   1  10   0   6   9  17  14  23   7   2  21   3

Davis Smith, Rows n = 1..144 of triangle, flattened
David G. Anderson, The Pi-Search Page.
D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, An Empirical Approach to the Normality of π, Experimental Math., 21 (2012), 375-384.
C. Sevcik, Fractal analysis of Pi normality, arXiv:1608.00430 [math.GM], 2016.
P. Trueb, Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi, arXiv:1612.00489 [math.NT], 2016.
Wikipedia, Normal number.

Cf. A000796, A022844, A066643, A068425, A176341.

Positions of 0 through 9 in base 10: A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008.

A332084_row(n)={my(L=List(vector(n,z,-1)), m=-1); while(vecmin(Vec(L))==-1, my(Z=lift(Mod(floor(Pi*n^(m++)),n))+1); if(L[Z]<0,listput(L,m,Z))); Vec(L)}

T(n,3) = 0, n > 3.

A100080 Position of first occurrence of n after the decimal point in the decimal expansion of 1/Pi.

Original entry on oeis.org

5, 2, 26, 1, 29, 19, 9, 13, 3, 6, 297, 64, 50, 385, 45, 18, 116, 65, 2, 41, 393, 102, 85, 125, 35, 93, 26, 86, 32, 43, 4, 1, 92, 58, 59, 69, 126, 12, 165, 151, 36, 717, 437, 196, 226, 29, 60, 160, 46, 55, 30, 112, 25, 19, 108, 90, 105, 134, 123, 70, 88, 9, 446, 149, 236, 511

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 03 2004

a(0) = A133268(1),
a(1) = A134251(1),
a(2) = A134252(1),
a(3) = A134253(1),
a(4) = A134254(1),
a(5) = A134255(1),
a(6) = A134256(1),
a(7) = A134257(1),
a(8) = A134258(1),
a(9) = A134259(1),
a(10) = A134260(1). - Artur Jasinski, Oct 16 2007

			1/Pi = 0.31830988618379067153776752674... so the first occurrence of 0 after the decimal point is at position 5; first occurrence of 1 is at position 2; first occurrence of 2 is at position 26; etc.

Cf. A032445 and A014777 for Pi, A078197 for e. Digits of 1/Pi=A049541.

Cf. A037000, A014777, A133268, A134251, A134252, A134253, A134254, A134255, A134256, A134257, A134258, A134259, A134260.

Table[ SequencePosition[#, IntegerDigits@ n][[1, 1]], {n, 0, 65}] &@ First@ RealDigits@ N[1/Pi, 10^4] (* James C. McMahon, Feb 06 2024 *)

Edited by N. J. A. Sloane, Jun 30 2008 at the suggestion of R. J. Mathar

A101196 Position of n-th n after the decimal point in Pi.

Original entry on oeis.org

1, 16, 17, 36, 48, 72, 96, 74, 55, 854, 709, 1080, 1076, 1636, 1657, 1651, 889, 1674, 1227, 2039, 1486, 2372, 2690, 2288, 2033, 2282, 1785, 2703, 4155, 3102, 3584, 3767, 4325, 3808, 3551, 4081, 3785, 3229, 4464, 4884, 4127, 4228, 5336, 3961, 4242, 3633

Offset: 1

Michael Joseph Halm, Dec 12 2004

			a(2) = 16 because the second occurrence of 2 in the digits of pi after its decimal point is at position 16, that is, after 141592653589793.

Cf. A014777, A037000, A037001, A037002, A037003, A037004, A037005, A037006, A037007, A037008.

Corrected and extended by Mark Hudson (mrmarkhudson(AT)hotmail.com), Dec 13 2004

cp's OEIS Frontend

A125628 Version of sexagesimal expansion of 2*Pi given by the Persian mathematician Al-Kashi in the 15th Century.

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A134210 Positions of 10 after the decimal point in the decimal expansion of Pi.

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A256521 Table T(n, k) of positions p[i] where number n occurs after the decimal point in the decimal expansion of Pi, read by antidiagonals.

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A332084 Triangle read by rows: T(n,k) is the smallest m >= 0 such that floor(Pi*n^m) == k (mod n), -1 if one does not exist, k = 0..n-1.

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A100080 Position of first occurrence of n after the decimal point in the decimal expansion of 1/Pi.

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A101196 Position of n-th n after the decimal point in Pi.

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