A280499 Triangular table for division in ring GF(2)[X]: T(n,k) = n/k, or 0 if k is not a divisor of n, where the binary expansion of each number defines the corresponding (0,1)-polynomial.

1, 2, 1, 3, 0, 1, 4, 2, 0, 1, 5, 0, 3, 0, 1, 6, 3, 2, 0, 0, 1, 7, 0, 0, 0, 0, 0, 1, 8, 4, 0, 2, 0, 0, 0, 1, 9, 0, 7, 0, 0, 0, 3, 0, 1, 10, 5, 6, 0, 2, 3, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 12, 6, 4, 3, 0, 2, 0, 0, 0, 0, 0, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 14, 7, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 15, 0, 5, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Jan 09 2017

Keywords

• nonn
• base
• tabl
• look

Comments

This is GF(2)[X] analog of A126988, using "carryless division in base-2" instead of ordinary division.

The triangular table T(n,k), n=1.., k=1..n is read by rows: T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), etc.

Examples

			The first 17 rows of the triangle:
   1
   2 1
   3 0 1
   4 2 0 1
   5 0 3 0 1
   6 3 2 0 0 1
   7 0 0 0 0 0 1
   8 4 0 2 0 0 0 1
   9 0 7 0 0 0 3 0 1
  10 5 6 0 2 3 0 0 0 1
  11 0 0 0 0 0 0 0 0 0 1
  12 6 4 3 0 2 0 0 0 0 0 1
  13 0 0 0 0 0 0 0 0 0 0 0 1
  14 7 0 0 0 0 2 0 0 0 0 0 0 1
  15 0 5 0 3 0 0 0 0 0 0 0 0 0 1
  16 8 0 4 0 0 0 2 0 0 0 0 0 0 0 1
  17 0 15 0 5 0 0 0 0 0 0 0 0 0 3 0 1
  -----------------------------------
7 ("111" in binary) encodes polynomial X^2 + X + 1, which is irreducible over GF(2) (7 is in A014580), so it is divisible only by itself and 1, and thus T(7,1) = 7, T(7,k) = 0 for k=2..6 and T(7,7) = 1.
9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), and thus T(9,3) = 7 and T(9,7) = 3 because the polynomial X + 1 is encoded by 3 ("11" in binary).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..32896; the first 256 rows of triangle
• Index entries for sequences related to binary expansion of n
• Index entries for sequences operating on polynomials in ring GF(2)[X]

Crossrefs

Lower triangular region of square array A280500.

Transpose: A280494.

Cf. A014580, A048720, A126988, A178908, A280500, A280493 (the row sums).

Programs

Scheme

(define (A280499 n) (A280500bi (A002024 n) (A002260 n))) ;; Code for A280500bi given in A280500.

Formula

T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.