|
%I #28 Jul 20 2021 07:01:46
%S 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 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 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 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 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 Conjecture: Pi is absolutely normal, meaning that -1 will never appear.
%C 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 Davis Smith, <a href="/A332084/b332084.txt">Rows n = 1..144 of triangle, flattened</a>
%H David G. Anderson, <a href="https://www.angio.net/pi/piquery">The Pi-Search Page</a>.
%H 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 C. Sevcik, <a href="https://arxiv.org/abs/1608.00430">Fractal analysis of Pi normality</a>, arXiv:1608.00430 [math.GM], 2016.
%H 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Normal_number">Normal number</a>.
%F T(n,3) = 0, n > 3.
%e The triangle T(n,k) starts:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 ...
%e 1: 0
%e 2: 1 0
%e 3: 0 2 4
%e 4: 1 3 2 0
%e 5: 1 7 3 0 8
%e 6: 1 9 14 0 10 2
%e 7: 1 7 10 0 8 6 2
%e 8: 3 1 8 0 9 6 14 5
%e 9: 10 1 2 0 3 20 18 11 5
%e 10: 32 1 6 0 2 4 7 13 11 5
%e 11: 5 1 22 0 13 4 2 6 9 24 12
%e 12: 5 1 10 0 3 17 14 18 2 6 20 10
%e 13: 5 1 10 0 6 9 17 14 23 7 2 21 3
%o (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 Cf. A000796, A022844, A066643, A068425, A176341.
%Y Positions of 0 through 9 in base 10: A037000, A037001, A037002, A037003, A037004, A037005, A036974, A037006, A037007, A037008.
%K nonn,tabl
%O 1,5
%A _Davis Smith_, Aug 22 2020
|
