A238595 Greatest common divisor of (x^m+y^m+(x+y)^m) - (z^m+t^m+(z+t)^m) over all x,y,z,t such that x^2 + xy + y^2 = z^2 + zt + t^2 and m=2n.

43200, 5644800, 10584000, 3801600, 706305600, 440294400, 2203200, 9116352000, 1327233600, 437184000, 210974400, 44689881600, 194184000, 223285708800, 12271089600, 652147200, 6448478400, 285538623552000, 837777600, 547348032000, 688766500800, 747187200

Offset

3,1

Comments

Provided that the quadruple (x,y,z,t) satisfies x^2+x*y+y^2 = z^2+z*t+t^2, define E(k,x,y,z,t) = (x^k+y^k+(x+y)^k) - (z^k+t^k+(z+t)^k). When k is odd, the only thing that can be said is that E(k,x,y,z,t) is even, and when k=4, E(k,x,y,z,t) equals 0. So the sequence will concern only k even and greater than 4, hence offset is 3 with even values m=2n only.

Links

Table of n, a(n) for n=3..24.

Étienne Ghys, Le plaisir de l’amateur, Images des Mathématiques, CNRS, 2009

Vincent Thill, Curiosités Arithmétiques, Editions Le Sanctuaire, 2008.

Example

The quadruple [5,6,1,9] satisfies (x^2+x*y+y^2) = (z^2+z*t+t^2). Then, for m=6, the expression is 302400 = 7*43200.

Prog

(PARI) q(x, y, n) = x^n+y^n+(x+y)^n;
expr(n, rv) = q(rv[1], rv[2], n) - q(rv[3], rv[4], n);
findgcd(n, v) = gcd(vector(#v, i, expr(n, v[i])));
lista(nn) = {vsols = [[5, 6, 1, 9], [4, 9, 1, 11], [7, 7, 2, 11], [8, 9, 3, 13], [7, 11, 3, 14], [5, 13, 2, 15], [9, 11, 4, 15], [8, 11, 1, 16], [10, 13, 5, 17], [7, 14, 1, 18]]; for (i=3, nn, print1(findgcd(2*i, vsols), ", "); ); }
\\ Michel Marcus, Mar 02 2014

Crossrefs

Sequence in context: A141085 A151623 A151660 * A205053 A146074 A321038

Adjacent sequences: A238592 A238593 A238594 * A238596 A238597 A238598

Keyword

nonn

Author

Michel Marcus, Mar 01 2014

Extensions

New name from Charles R Greathouse IV, Mar 03 2014

Status

approved