A037005 -id:A037005 - cp's OEIS frontend

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.

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

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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