A037001 -id:A037001 - cp's OEIS frontend

A083609 Starting positions of strings of six 2's in the decimal expansion of Pi.

963024, 1637080, 1795773, 2523356, 3474036, 5463417, 5803105, 7024615, 9742967, 11836401, 12883291, 13208202, 13371031, 15419528, 15783557, 18183625, 19081176, 20031349, 20363606, 20399387, 20735063, 21682696, 25303344, 31104717, 31614606, 32300569, 33093853, 34422277

Offset: 1

Rick L. Shepherd, May 01 2003

Dave Andersen, Pi-Search Page
Eric Weisstein's World of Mathematics, Pi Digits.
Index entries for sequences related to the number Pi

Cf. A083608 (five "2"s), A118079 (seven "2"s); A037001 = A053746 - 1 (any "2"s).

With[{s = ConstantArray[2, 6]}, SequencePosition[First@ RealDigits@ N[Pi, 10^8], s][[All, 1]] - 1] (* Michael De Vlieger, Mar 20 2017, Version 10.1 *)

More terms from Jinyuan Wang, Feb 29 2020

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.

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

A083609 Starting positions of strings of six 2's in the decimal expansion of Pi.

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

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