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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.

%I A332084 #28 Jul 20 2021 07:01:46
%S A332084 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,
%T A332084 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,
%U A332084 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
%N 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.
%C A332084 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.
%C A332084 Conjecture: Pi is absolutely normal, meaning that -1 will never appear.
%C A332084 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.
%H A332084 Davis Smith, <a href="/A332084/b332084.txt">Rows n = 1..144 of triangle, flattened</a>
%H A332084 David G. Anderson, <a href="https://www.angio.net/pi/piquery">The Pi-Search Page</a>.
%H A332084 D. H. Bailey, J. M. Borwein, C. S. Calude, M. J. Dinneen, M. Dumitrescu, and A. Yee, <a href="https://doi.org/10.1080/10586458.2012.665333">An Empirical Approach to the Normality of π</a>, Experimental Math., 21 (2012), 375-384.
%H A332084 C. Sevcik, <a href="https://arxiv.org/abs/1608.00430">Fractal analysis of Pi normality</a>, arXiv:1608.00430 [math.GM], 2016.
%H A332084 P. Trueb, <a href="https://arxiv.org/abs/1612.00489">Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi</a>, arXiv:1612.00489 [math.NT], 2016.
%H A332084 Wikipedia, <a href="https://en.wikipedia.org/wiki/Normal_number">Normal number</a>.
%F A332084 T(n,3) = 0, n > 3.
%e A332084 The triangle T(n,k) starts:
%e A332084 n\k   0   1   2   3   4   5   6   7   8   9  10  11  12 ...
%e A332084 1:    0
%e A332084 2:    1   0
%e A332084 3:    0   2   4
%e A332084 4:    1   3   2   0
%e A332084 5:    1   7   3   0   8
%e A332084 6:    1   9  14   0  10   2
%e A332084 7:    1   7  10   0   8   6   2
%e A332084 8:    3   1   8   0   9   6  14   5
%e A332084 9:   10   1   2   0   3  20  18  11   5
%e A332084 10:  32   1   6   0   2   4   7  13  11   5
%e A332084 11:   5   1  22   0  13   4   2   6   9  24  12
%e A332084 12:   5   1  10   0   3  17  14  18   2   6  20  10
%e A332084 13:   5   1  10   0   6   9  17  14  23   7   2  21   3
%o A332084 (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)}
%Y A332084 Cf. A000796, A022844, A066643, A068425, A176341.
%Y A332084 Positions of 0 through 9 in base 10: A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008.
%K A332084 nonn,tabl
%O A332084 1,5
%A A332084 _Davis Smith_, Aug 22 2020