A386377 a(n) is the number of solutions to the equation x^2 + y^3 + z^4 = w^5 where GCD(x, y, z)=1.

0, 1, 0, 0, 0, 0, 2, 2, 5, 0, 1, 1, 0, 1, 1, 1, 3, 2, 1, 2, 2, 0, 2, 2, 4, 1, 0, 2, 2, 1, 2, 1, 13, 0, 2, 0, 1, 3, 1, 1, 4, 0, 0, 7, 5, 3, 0, 2, 10, 1, 1, 2, 7, 2, 1, 1, 8, 1, 2, 1, 7, 0, 4, 3, 8, 4, 4, 1, 1, 5, 1, 0, 11, 1, 2, 0, 3, 1, 3, 5, 12, 7, 2, 2, 2, 2, 0, 1, 14, 2, 2, 1

Offset: 1

Author

David A. Corneth and Zhining Yang, Jul 20 2025

Keywords

• nonn

Examples

			a(9) = 5 because x^2 + y^3 + z^4 = 9^5 where GCD(x,y,z)=1 has 5 positive integer solutions :{220,22,1},{64,38,3},{241,7,5},{9,38,8},{118,29,12}.
		

Links

• Zhining Yang, Table of n, a(n) for n = 1..4500 (terms 1..856 from David A. Corneth)
• David A. Corneth, Tuples (x, y, z, w) that are solutions to the equation.
• David A. Corneth, PARI program

Crossrefs

Cf. A386373, A386521.

Programs

Mathematica

f[w_]:=(c=0;zz=w^5;Do[yy=zz-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,c++]],{y,Floor[yy^(1/3)]}],{z,Floor[zz^(1/4)]}];c);Array[f@#&, 30]