A357749 Sorted list of nonzero numbers x, y, z that occur in solutions to the equation (x + y)^2 + (y + z)^2 + (z + x)^2 = 12*x*y*z.

1, 3, 13, 61, 217, 291, 1393, 3673, 4683, 6673, 16693, 31971, 62221, 106153, 153181, 360517, 733933, 1054081, 1285131, 1709221, 2430493, 3516483, 4778353, 16848481, 17857153, 21717363, 27755113, 38745493, 55764867, 80725921, 98938381, 185236633, 302517517, 386781123

Offset: 1

Author

Hugo Pfoertner, Oct 18 2022

Keywords

• nonn

Examples

			The first few solutions are (x, y, z) = (1, 1, 1), (1, 1, 3), (1, 3, 13), (1, 13, 61), (1, 61, 291), (1, 291, 1393), (1, 1393, 6673), (1, 6673, 31971), (3, 13, 217), (3, 217, 3673), ..., so 1, 3, 13, 61, 217, 291, ... are terms.
		

Links

• Yasuaki Gyoda, Positive integer solutions to (x+y)^2+(y+z)^2+(z+x)^2=12xyz, arXiv:2109.09639 [math.NT], 2022.

Crossrefs

Cf. A002559, A101368, A357870.

Programs

PARI

a357749(steps) = {L=List(); listput(L,[1,1,1]); listput(L,[1,1,3]); listput(L,[1,13,3]); for(n=1,steps,my(mp,mv);for(l=1,#L,mv=vecmax(L[l],&mp); my (a=L[l][1], b=L[l][2], c=L[l][3], s=a+b+c); if(mp==1,listput(L,[a,6*a*c-s,c]);listput(L,[a,b,6*a*b-s])); if(mp==2,listput(L,[6*b*c-s,b,c]);listput(L,[a,b,6*a*b-s])); if(mp==3,listput(L,[6*b*c-s,b,c]);listput(L,[a,6*a*c-s,c]))));M=List(); for (k=1, #L, for(j=1, 3, listput(M,L[k][j]))); vecsort(M,,8)};
a357749(13)[1..35]