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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.

%I A280499 #28 Jan 11 2017 09:06:54
%S A280499 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,
%T A280499 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,
%U A280499 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
%N 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.
%C A280499 This is GF(2)[X] analog of A126988, using "carryless division in base-2" instead of ordinary division.
%C A280499 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.
%H A280499 Antti Karttunen, <a href="/A280499/b280499.txt">Table of n, a(n) for n = 1..32896; the first 256 rows of triangle</a>
%H A280499 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A280499 <a href="/index/Ge#GF2X">Index entries for sequences operating on polynomials in ring GF(2)[X]</a>
%F A280499 T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.
%e A280499 The first 17 rows of the triangle:
%e A280499    1
%e A280499    2 1
%e A280499    3 0 1
%e A280499    4 2 0 1
%e A280499    5 0 3 0 1
%e A280499    6 3 2 0 0 1
%e A280499    7 0 0 0 0 0 1
%e A280499    8 4 0 2 0 0 0 1
%e A280499    9 0 7 0 0 0 3 0 1
%e A280499   10 5 6 0 2 3 0 0 0 1
%e A280499   11 0 0 0 0 0 0 0 0 0 1
%e A280499   12 6 4 3 0 2 0 0 0 0 0 1
%e A280499   13 0 0 0 0 0 0 0 0 0 0 0 1
%e A280499   14 7 0 0 0 0 2 0 0 0 0 0 0 1
%e A280499   15 0 5 0 3 0 0 0 0 0 0 0 0 0 1
%e A280499   16 8 0 4 0 0 0 2 0 0 0 0 0 0 0 1
%e A280499   17 0 15 0 5 0 0 0 0 0 0 0 0 0 3 0 1
%e A280499   -----------------------------------
%e A280499 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.
%e A280499 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).
%o A280499 (Scheme) (define (A280499 n) (A280500bi (A002024 n) (A002260 n))) ;; Code for A280500bi given in A280500.
%Y A280499 Lower triangular region of square array A280500.
%Y A280499 Transpose: A280494.
%Y A280499 Cf. A014580, A048720, A126988, A178908, A280500, A280493 (the row sums).
%K A280499 nonn,base,tabl,look
%O A280499 1,2
%A A280499 _Antti Karttunen_, Jan 09 2017