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A147696 Triangle read by rows: numbers n and columns k such that T(n, k) is n mod k.

%I A147696 #8 Feb 16 2025 08:33:09
%S A147696 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 A147696 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 A147696 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 A147696 Triangle read by rows: numbers n and columns k such that T(n, k) is n mod k.
%C A147696 The triangle begins with (2, 2).
%C A147696 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 A147696 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 A147696 Zeros correspond to ones in the Redheffer matrix. Various interesting patterns exist. For example, as noted above, T(n^2, n) = 0. Also:
%C A147696 T(n^2 + n, n) = T(n^2 + n, n + 1) = 0
%C A147696 T(n^2 + n - 2, n - 1) = 0
%C A147696 T(n^2 - 1, n - 1) = 0
%C A147696 For all k in some [0, c]:
%C A147696 T(n^2, 2 + k) = 0 if n is even
%C A147696 T(n^2, 2 + k) = 1 if n is odd
%C A147696 T(n^2 + n, 2 + k) = 0
%C A147696 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 A147696 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RedhefferMatrix.html">Redheffer Matrix</a>
%e A147696 The triangle begins:
%e A147696 0
%e A147696 1
%e A147696 0 1
%e A147696 1 2
%e A147696 0 0
%e A147696 1 1
%e A147696 0 2
%e A147696 1 0 1
%e A147696 0 1 2
%e A147696 1 2 3
%e A147696 0 0 0
%e A147696 1 1 1
%e A147696 0 2 2
%e A147696 1 0 3
%e A147696 0 1 0 1
%e A147696 1 2 1 2
%e A147696 0 0 2 3
%e A147696 1 1 3 4
%e A147696 0 2 0 0
%e A147696 1 0 1 1
%e A147696 0 1 2 2
%e A147696 1 2 3 3
%e A147696 0 0 0 4
%Y A147696 Cf. A002321, A083058, A144912
%K A147696 easy,nonn,tabf
%O A147696 2,6
%A A147696 _Reikku Kulon_, Nov 10 2008