A106449 - cp's OEIS frontend

A106449 Square array (P(x) XOR P(y))/gcd(P(x),P(y)) where P(x) and P(y) are polynomials with coefficients in {0,1} given by the binary expansions of x and y, and all calculations are done in polynomial ring GF(2)[X], with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by antidiagonals.

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

Offset: 1

Antti Karttunen, May 21 2005

Array is read by antidiagonals, with row x and column y ranging as: (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
"Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1).
This is GF(2)[X] analog of A106448. In the definition XOR means addition in polynomial ring GF(2)[X], that is, a carryless binary addition, A003987.

			The top left 17 X 17 corner of the array:
        1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17
     +--------------------------------------------------------------------
   1 :  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, 12, 15, 14, 17, 16, ...
   2 :  3,  0,  1,  3,  7,  2,  5,  5, 11,  4,  9,  7, 15,  6, 13,  9, 19, ...
   3 :  2,  1,  0,  7,  2,  3,  4, 11,  6,  7,  8,  5, 14, 13,  4, 19, 14, ...
   4 :  5,  3,  7,  0,  1,  1,  3,  3, 13,  7, 15,  2,  9,  5, 11,  5, 21, ...
   5 :  4,  7,  2,  1,  0,  1,  2, 13,  4,  3, 14,  7,  8, 11,  2, 21,  4, ...
   6 :  7,  2,  3,  1,  1,  0,  1,  7,  5,  2, 13,  3, 11,  4,  7, 11, 13, ...
   7 :  6,  5,  4,  3,  2,  1,  0, 15,  2, 13, 12, 11, 10,  3,  8, 23, 22, ...
   8 :  9,  5, 11,  3, 13,  7, 15,  0,  1,  1,  3,  1,  5,  3,  7,  3, 25, ...
   9 :  8, 11,  6, 13,  4,  5,  2,  1,  0,  1,  2,  3,  4,  1,  2, 25,  8, ...
  10 : 11,  4,  7,  7,  3,  2, 13,  1,  1,  0,  1,  1,  7,  2,  1, 13,  7, ...
  11 : 10,  9,  8, 15, 14, 13, 12,  3,  2,  1,  0,  7,  6,  5,  4, 27, 26, ...
  12 : 13,  7,  5,  2,  7,  3, 11,  1,  3,  1,  7,  0,  1,  1,  1,  7, 11, ...
  13 : 12, 15, 14,  9,  8, 11, 10,  5,  4,  7,  6,  1,  0,  3,  2, 29, 28, ...
  14 : 15,  6, 13,  5, 11,  4,  3,  3,  1,  2,  5,  1,  3,  0,  1, 15, 31, ...
  15 : 14, 13,  4, 11,  2,  7,  8,  7,  2,  1,  4,  1,  2,  1,  0, 31,  2, ...
  16 : 17,  9, 19,  5  21, 11, 23,  3, 25, 13, 27,  7, 29, 15, 31,  0,  1, ...
  17 : 16, 19, 14, 21,  4, 13, 22, 25,  8,  7, 26, 11, 28, 31,  2,  1,  0, ...

Row 1: A004442 (without its initial term), row 2: A106450 (without its initial term).

Cf. A003987, A048720, A091255, A106448, A280500.

up_to = 105;
A106449sq(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); fromdigits(Vec(lift((Pa+Pb)/gcd(Pa,Pb))),2); }; \\ Note that XOR is just + in GF(2)[X] world.
A106449list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A106449sq(col,(a-(col-1))))); (v); };
v106449 = A106449list(up_to);
A106449(n) = v106449[n]; \\ Antti Karttunen, Oct 21 2019

A(x, y) = A280500(A003987(x, y), A091255(x, y)), that is, A003987(x, y) = A048720(A(x, y), A091255(x, y)).

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