A386521 Integers w such that the Diophantine equation x^2 + y^3 + z^4 = w^5 with GCD(x,y,z)=1 has no positive integer solutions.

1, 3, 4, 5, 6, 10, 13, 22, 27, 34, 36, 42, 43, 47, 62, 72, 76, 87, 95, 102, 111, 183, 251, 279, 315, 322, 327, 344, 483, 490, 528, 615, 708, 762, 1170, 1302, 2295, 2526, 3282, 3382, 6012

Offset: 1

Author

David A. Corneth and Zhining Yang, Jul 24 2025

Keywords

• nonn
• more

Comments

a(42) > 6500. - Giovanni Resta, Aug 12 2025

Examples

			9 is not a term because 9^5 = x^2 + y^3 + z^4 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}.
		

Crossrefs

Cf. A386373, A386377.

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);Select[Range@50,f@#==0&]

Extensions

a(41) from Giovanni Resta, Aug 12 2025