login
A372727
Triangle read by rows: T(n, k) = n if k = 0, otherwise n - k*floor(n/k). The binary modulo operation.
1
0, 1, 0, 2, 0, 0, 3, 0, 1, 0, 4, 0, 0, 1, 0, 5, 0, 1, 2, 1, 0, 6, 0, 0, 0, 2, 1, 0, 7, 0, 1, 1, 3, 2, 1, 0, 8, 0, 0, 2, 0, 3, 2, 1, 0, 9, 0, 1, 0, 1, 4, 3, 2, 1, 0, 10, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0, 12, 0, 0, 0, 0, 2, 0, 5, 4, 3, 2, 1, 0
OFFSET
0,4
COMMENTS
The binary operation 'mod' as defined here is discussed in 'Concrete Mathematics' by Graham et. al. on p. 82 and the connection with the congruence relation '(mod)' on p. 123. See also Bach & Shallit, p. 21, and Apostol, p. 14.
This definition is implemented in Sage, but not in Python. For example, Sage answers 0.mod(0) = 0, whereas in Python 0 % 0 leads to a 'ZeroDivisionError'. What is often misunderstood is that the operation 'mod' gives answers to divisibility, not to division. Apostol shows that n|0 (every integer divides zero), but 0|n implies n = 0 (zero divides only zero), and thus confirms the result given by Sage.
REFERENCES
Tom Apostol, Introduction to analytic number theory, 1976, Springer, page 14.
Eric Bach and Jeffrey Shallit, Algorithmic Number Theory, 1997, p. 21.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, 34th printing 2022, p. 81f.
EXAMPLE
Triangle begins:
[ 0] 0;
[ 1] 1, 0;
[ 2] 2, 0, 0;
[ 3] 3, 0, 1, 0;
[ 4] 4, 0, 0, 1, 0;
[ 5] 5, 0, 1, 2, 1, 0;
[ 6] 6, 0, 0, 0, 2, 1, 0;
[ 7] 7, 0, 1, 1, 3, 2, 1, 0;
[ 8] 8, 0, 0, 2, 0, 3, 2, 1, 0;
[ 9] 9, 0, 1, 0, 1, 4, 3, 2, 1, 0;
[10] 10, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0;
[11] 11, 0, 1, 2, 3, 1, 5, 4, 3, 2, 1, 0;
.
The triangle shows the modulo operation in the range 0 <= k <= n. Test your
computer implementation in the range R X R where R = [-6, ..., 0, ..., 6].
According to Graham et al. it should look like this:
0, -1, -2, 0, 0, 0, -6, 0, 0, 0, 2, 4, 0
-5, 0, -1, -2, -1, 0, -5, 0, 1, 1, 3, 0, 1
-4, -4, 0, -1, 0, 0, -4, 0, 0, 2, 0, 1, 2
-3, -3, -3, 0, -1, 0, -3, 0, 1, 0, 1, 2, 3
-2, -2, -2, -2, 0, 0, -2, 0, 0, 1, 2, 3, 4
-1, -1, -1, -1, -1, 0, -1, 0, 1, 2, 3, 4, 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
-5, -4, -3, -2, -1, 0, 1, 0, 1, 1, 1, 1, 1
-4, -3, -2, -1, 0, 0, 2, 0, 0, 2, 2, 2, 2
-3, -2, -1, 0, -1, 0, 3, 0, 1, 0, 3, 3, 3
-2, -1, 0, -2, 0, 0, 4, 0, 0, 1, 0, 4, 4
-1, 0, -3, -1, -1, 0, 5, 0, 1, 2, 1, 0, 5
0, -4, -2, 0, 0, 0, 6, 0, 0, 0, 2, 1, 0
MAPLE
MOD := (n, k) -> ifelse(k = 0, n, n - k * iquo(n, k)):
seq( seq(MOD(n, k), k = 0..n), n = 0..12);
PROG
(SageMath)
def T(n, k): return n.mod(k)
for n in srange(12): print([T(n, k) for k in range(n + 1)])
(Python)
def T(n, k): return n if k == 0 else n - k * (n // k)
for n in range(12): print([T(n, k) for k in range(n + 1)])
(Python)
def A372727_T(n, k): return n % k if k else n # Chai Wah Wu, May 14 2024
CROSSREFS
Cf. A111490 (row sums).
Cf. A048158.
Sequence in context: A212163 A212195 A228926 * A321414 A268865 A024159
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 13 2024
STATUS
approved