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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
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
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
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
KEYWORD
nonn,tabl
AUTHOR
Davis Smith, Aug 22 2020
STATUS
approved