

A147696


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


0



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
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OFFSET

2,6


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.
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

Table of n, a(n) for n=2..105.
Eric Weisstein's World of Mathematics, Redheffer Matrix


CROSSREFS

Cf. A002321, A083058, A144912
Sequence in context: A323069 A325334 A280287 * A001842 A216654 A326016
Adjacent sequences: A147693 A147694 A147695 * A147697 A147698 A147699


KEYWORD

easy,nonn,tabf


AUTHOR

Reikku Kulon, Nov 10 2008


STATUS

approved



