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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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.

LINKS

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.

FORMULA

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

EXAMPLE

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

PROG

(PARI) 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)}

CROSSREFS

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

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

Sequence in context: A190555 A141843 A322169 * A130266 A261595 A211197

Adjacent sequences:  A332081 A332082 A332083 * A332085 A332086 A332087

KEYWORD

nonn,tabl,changed

AUTHOR

Davis Smith, Aug 22 2020

STATUS

approved

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Last modified July 24 05:23 EDT 2021. Contains 346273 sequences. (Running on oeis4.)