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

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, 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, 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

Offset: 2

Author

Reikku Kulon, Nov 10 2008

Keywords

• easy
• nonn
• tabf

Comments

The triangle begins with (2, 2).

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.

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.

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

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

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

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

For all k in some [0, c]:

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

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

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

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.

Examples

			The triangle begins:
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 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 4
		

Links

• Eric Weisstein's World of Mathematics, Redheffer Matrix

Crossrefs

Cf. A002321, A083058, A144912