login
Triangle read by rows: numbers n and columns k such that T(n, k) is n mod k.
0

%I #7 Oct 31 2021 13:53:09

%S 0,1,0,1,1,2,0,0,1,1,0,2,1,0,1,0,1,2,1,2,3,0,0,0,1,1,1,0,2,2,1,0,3,0,

%T 1,0,1,1,2,1,2,0,0,2,3,1,1,3,4,0,2,0,0,1,0,1,1,0,1,2,2,1,2,3,3,0,0,0,

%U 4,1,1,1,0,1,0,2,2,1,2,1,0,3,2,3,0,1,0,3,4,1,2,1,4,5,0,0,2,0,0,1,1,3,1,1

%N Triangle read by rows: numbers n and columns k such that T(n, k) is n mod k.

%C The triangle begins with (2, 2).

%C Each row can be produced from the previous row by adding one to each number and resetting to zero any which would equal their column number. A number p > 2 is prime iff row p contains no zeros.

%C A new column k begins at row n when n is a perfect square. T(n, k) is then 1, while T(n, sqrt(n) = k - 1) is 0.

%C Zeros correspond to ones in the Redheffer matrix. Various interesting patterns exist. For example, as noted above, T(n^2, n) = 0. Also:

%C T(n^2 + n, n) = T(n^2 + n, n + 1) = 0

%C T(n^2 + n - 2, n - 1) = 0

%C T(n^2 - 1, n - 1) = 0

%C For all k in some [0, c]:

%C T(n^2, 2 + k) = 0 if n is even

%C T(n^2, 2 + k) = 1 if n is odd

%C T(n^2 + n, 2 + k) = 0

%C Every zero is located on some parabola directed toward n = 0, having either even width and produced by an even sequence; or having an odd width and produced by an odd sequence. In either case, the relevant sequence has constant first differences 2. T(n^2, n) begins an odd parabola, while T(n^2 + n, n) begins an even parabola and parabolas of either variety extend from infinitely many other locations.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RedhefferMatrix.html">Redheffer Matrix</a>

%e The triangle begins:

%e 0

%e 1

%e 0 1

%e 1 2

%e 0 0

%e 1 1

%e 0 2

%e 1 0 1

%e 0 1 2

%e 1 2 3

%e 0 0 0

%e 1 1 1

%e 0 2 2

%e 1 0 3

%e 0 1 0 1

%e 1 2 1 2

%e 0 0 2 3

%e 1 1 3 4

%e 0 2 0 0

%e 1 0 1 1

%e 0 1 2 2

%e 1 2 3 3

%e 0 0 0 4

%Y Cf. A002321, A083058, A144912

%K easy,nonn,tabf

%O 2,6

%A _Reikku Kulon_, Nov 10 2008