A325825 Square array giving the monic polynomial q satisfying q = gcd(P(x),P(y)) where P(x) and P(y) are polynomials in ring GF(3)[X] with coefficients in {0,1,2} given by the ternary expansions of x and y. The polynomial q is converted back to a ternary number, and then expressed in decimal.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 5, 3, 5, 4, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 3, 1, 1, 3, 1, 4, 3, 1, 1

Offset: 1

Author

Antti Karttunen, May 22 2019

Keywords

• nonn
• tabl

Comments

Array is symmetric, and is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...

If there is a polynomial q that satisfies q = gcd(P(x),P(y)), then also polynomial -q (which is obtained by changing all nonzero coefficients of q as 1 <--> 2, see A004488) satisfies the same relation, because there are two units (+1 and -1) in polynomial ring GF(3)[X]. Here we always choose the polynomial that is monic (i.e., with a leading coefficient +1), thus its base-3 encoding has "1" as its most significant digit, and the terms given here are all included in A132141.

Examples

			The array begins as:
   y
x      1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  12,  ...
   --+-----------------------------------------------------
   1 | 1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1,  1,  ...
   2 | 1,  1,  1,  1,  1,  1,  1,  1,  1,   1,  1,  1,  ...
   3 | 1,  1,  3,  1,  1,  3,  1,  1,  3,   1,  1,  3,  ...
   4 | 1,  1,  1,  4,  1,  1,  1,  4,  1,   1,  4,  4,  ...
   5 | 1,  1,  1,  1,  5,  1,  5,  1,  1,   1,  5,  1,  ...
   6 | 1,  1,  3,  1,  1,  3,  1,  1,  3,   1,  1,  3,  ...
   7 | 1,  1,  1,  1,  5,  1,  5,  1,  1,   1,  5,  1,  ...
   8 | 1,  1,  1,  4,  1,  1,  1,  4,  1,   1,  4,  4,  ...
   9 | 1,  1,  3,  1,  1,  3,  1,  1,  9,   1,  1,  3,  ...
  10 | 1,  1,  1,  1,  1,  1,  1,  1,  1,  10,  1,  1,  ...
  11 | 1,  1,  1,  4,  5,  1,  5,  4,  1,   1, 11,  4,  ...
  12 | 1,  1,  3,  4,  1,  3,  1,  4,  3,   1,  4, 12,  ...
		

Crossrefs

Cf. A004488, A091255, A132141, A325820, A325821, A325827.

Central diagonal: A330740 (after its initial zero).

Programs

PARI

up_to = 105;
A004488(n) = subst(Pol(apply(x->(3-x)%3, digits(n, 3)), 'x), 'x, 3);
A325825sq(a,b) = { my(a=fromdigits(Vec(lift(gcd(Pol(digits(a,3))*Mod(1, 3),Pol(digits(b,3))*Mod(1, 3)))),3), b=A004488(a)); min(a,b); };
A325825list(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] = A325825sq(col,(a-(col-1))))); (v); };
v325825 = A325825list(up_to);
A325825(n) = v325825[n];