A380305 Triangle read by rows: T(n,k) = (n - floor((n - k)/k)) mod k, for 0 < k <= n.

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 2, 1, 0, 0, 1, 0, 3, 2, 1, 0, 0, 1, 1, 3, 3, 2, 1, 0, 0, 0, 1, 0, 4, 3, 2, 1, 0, 0, 0, 2, 1, 4, 4, 3, 2, 1, 0, 0, 1, 0, 2, 0, 5, 4, 3, 2, 1, 0, 0, 1, 0, 2, 1, 5, 5, 4, 3, 2, 1, 0, 0, 0, 1, 3, 2, 0, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Author

Lechoslaw Ratajczak, Jan 19 2025

Keywords

• nonn
• tabl

Comments

The triangle is a variant of the triangle in A048158. The period of the k-th column consists of the period of the k-th column in the triangle in A048158 (0,1,2,...,k-1) and its consecutive k-1 right rotations (moving the rightmost element to the left end). Thus the k-th column has period k^2 and the r-th element of this period has the form (r - 1 - floor((r - 1)/k)) mod k (1 <= r <= k^2).

Such as the triangle in A048158 may be the basis for definitions of different kinds of numbers (abundant numbers, perfect numbers, etc.), this triangle may be the basis for definitions of counterparts of these numbers (elements of A375595 as counterparts of abundant numbers, elements of A380153 as counterparts of perfect numbers).

Examples

			Triangle begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 ...
----------------------------
  1| 0
  2| 0 0
  3| 0 1 0
  4| 0 1 1 0
  5| 0 0 2 1 0
  6| 0 0 2 2 1 0
  7| 0 1 0 3 2 1 0
  8| 0 1 1 3 3 2 1 0
  9| 0 0 1 0 4 3 2 1 0
 10| 0 0 2 1 4 4 3 2 1  0
 11| 0 1 0 2 0 5 4 3 2  1  0
 ...
		

Crossrefs

Cf. A048158, A375595, A378275, A380153.

Programs

Mathematica

T[n_,k_]:=Mod[n - Floor[(n - k)/k], k]; Table[T[n,k], {n,13},{k,n}]//Flatten (* Stefano Spezia, Jan 20 2025 *)

Maxima

(for n:1 thru 25 do
(for k:1 thru n do
(T[n,k]:mod(n-floor((n-k)/k),k)),
print(makelist(T[n,i], i, 1, n))));

PARI

row(n) = vector(n, k, (n - floor((n - k)/k)) % k); \\ Michel Marcus, Jan 20 2025