A037008 -id:A037008 - cp's OEIS frontend

A134219 Positions of 19 after decimal point in decimal expansion of Pi.

37, 168, 198, 246, 390, 417, 432, 495, 541, 704, 717, 843, 945, 975, 985, 997, 1047, 1166, 1227, 1237, 1345, 1384, 1427, 1535, 1618, 1641, 1733, 1881, 1915, 1944, 2054, 2128, 2821, 2856, 2872, 2897, 2902, 2905, 2918, 2944, 2960, 2997, 3030, 3166, 3337, 3358

Offset: 1

Artur Jasinski, Oct 14 2007

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

Cf. A037008, A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A014777, A050208, A134212, A134213, A134214, A134215, A134216, A134217, A134218.

Transpose[SequencePosition[RealDigits[Pi,10,10000][[1]],{1,9}]][[1]]-1 (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, May 09 2016 *)

A014974 Differences between successive locations of zeros in decimal expansion of Pi.

Original entry on oeis.org

18, 4, 11, 6, 6, 8, 12, 9, 10, 5, 7, 4, 14, 13, 5, 3, 9, 19, 12, 38, 3, 16, 6, 17, 4, 16, 1, 3, 16, 3, 10, 17, 3, 1, 5, 3, 6, 23, 5, 5, 13, 22, 8, 42, 20, 7, 3, 20, 2, 7, 5, 4, 35, 5, 1, 1, 15, 20, 19, 2, 10, 13, 2, 19, 12, 5

Offset: 1

Bagirath R. Krishnamachari (bagi(AT)callisto.miel.mot.com)

Assuming Pi is normal, this sequence includes every finite sequence of positive integers. - Franklin T. Adams-Watters, Mar 15 2006

			First two 0's are in 33rd and 51st digits, difference is 18.

Harvey P. Dale, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Normal Number.

Cf. A000796, A037008, A139728, A108664.

Differences[Flatten[Position[RealDigits[Pi,10,1000][[1]],0]]] (* Harvey P. Dale, Jul 04 2017 *)

a(n) = A014976(n+1) - A014976(n). - Michel Marcus, Oct 06 2013

More terms from Simon Plouffe.

A133268 a(n) = positions of 0's after decimal point in decimal expansion of 1/Pi.

Original entry on oeis.org

5, 15, 31, 37, 48, 79, 81, 84, 89, 95, 118, 137, 189, 222, 232, 240, 258, 264, 269, 279, 298, 314, 315, 362, 371, 394, 435, 451, 460, 463, 466, 472, 480, 497, 507, 510, 520, 521, 525, 541, 565, 569, 571, 596, 600, 606, 609, 610, 636, 670, 702, 703, 706, 707

Offset: 1

Artur Jasinski, Oct 16 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Cf. A037008, A049541, A014777.

Flatten[Position[RealDigits[1/Pi,10,1000][[1]],0]] (* Harvey P. Dale, May 16 2012 *)

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.

A383732 a(n) is the smallest k such that every digit from 0 to 9 appears at least n times among the first k digits of Pi (after the decimal point).

Original entry on oeis.org

32, 50, 54, 65, 71, 77, 96, 99, 120, 139, 156, 166, 209, 224, 232, 235, 242, 288, 299, 301, 306, 320, 343, 351, 405, 407, 412, 429, 439, 452, 458, 463, 468, 475, 478, 486, 506, 538, 540, 544, 548, 556, 559, 560, 567, 569, 575, 577, 584, 591, 609, 621, 622, 625, 626, 631, 633, 634, 641

Offset: 1

Guy Amit, May 07 2025

The first 6 terms also appear in A037008, the position of the digit 0 in the decimal expansion of Pi.
From Pontus von Brömssen, May 13 2025: (Start)
If the digit "3" before the decimal point is included (but a(n) still represents the number of digits after the decimal point), the first difference to this sequence (i.e., the first time the extra "3" is useful) would be a(229) = 2583 instead of 2597.
For d = 0, 1, ..., 9, the smallest n >= 1 for which d is the last digit to occur n times is 1, 146359, 2100, 229, 94, 61, 118, 7, 794, 9734, respectively.
(End)

			For n = 1, a(1) = 32 since this is the first position in the decimal expansion of Pi such that every digit from 0 to 9 has appeared at least once among the first 32 digits.
For n = 2, a(2) = 50 since this is the first position in the decimal expansion of Pi such that every digit from 0 to 9 has appeared at least twice among the first 50 digits.

Guy Amit, Table of n, a(n) for n = 1..10000

Cf. A000796, A037008.

% Assuming x contains the digits of Pi, x = [1, 4, 1, 5, 9, ...]
a = nan(1);
counts = zeros(10, 1);
n = 1;
for i = 1:length(x)
    for j = 1:10
        if x(i) == (j-1)
            counts(j) = counts(j) + 1;
        end
    end
    if sum(counts>=n) == 10
        a(n) = i;
        n = n + 1;
    end
end

piDigit = Rest[RealDigits[Pi, 10, 700][[1]]];
f[pd_List]:=Module[{res,len,count,n=1},count=CreateDataStructure["FixedArray",ConstantArray[0,10]];res=CreateDataStructure["Queue"];len=Length[pd]; Do[Do[If[pd[[i]]==j-1,count["SetPart",j,1+count["Part",j]]],{j,10}];If[Total[Boole[#>=n]&/@(count["Elements"])]==10,res["Push",i];n++],{i,len}];res["Elements"]];f[piDigit] (* Shenghui Yang, May 19 2025 *) (* or *)
upto[nd_] := Block[{pi=Rest@ RealDigits[Pi,10,nd][[1]], cnt=0 Range[10], n=1}, Reap[Do[cnt[[pi[[j]] + 1]]++; If[Min[cnt-n] == 0, n++; Sow@j], {j, Length@ pi}]][[2,1]]]; upto[700] (* for older Mma, Giovanni Resta, May 22 2025 *)

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

A134219 Positions of 19 after decimal point in decimal expansion of Pi.

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A014974 Differences between successive locations of zeros in decimal expansion of Pi.

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A133268 a(n) = positions of 0's after decimal point in decimal expansion of 1/Pi.

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Mathematica

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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A383732 a(n) is the smallest k such that every digit from 0 to 9 appears at least n times among the first k digits of Pi (after the decimal point).

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

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