A386373 a(n) is the smallest integer w such that the equation x^2 + y^3 + z^4 = w^5 where GCD(x,y,z)=1 has exactly n positive integer solutions.

2, 7, 17, 25, 9, 100, 44, 57, 117, 49, 73, 81, 33, 89, 177, 193, 305, 161, 257, 273, 425, 289, 697, 441, 313, 689, 369, 593, 809, 233, 761, 1865, 2001, 857, 1121, 649, 1353, 865, 521, 1257, 577, 681, 2081, 1409, 1169, 1753, 1801, 1201, 1745, 2833, 3853, 3649, 3353, 1305, 793

Offset: 1

Author

Zhining Yang, Jul 19 2025

Keywords

• nonn

Comments

From David A. Corneth, Jul 20 2025: (Start)

a(41) = 577. If a(41) is 1 (mod 8) then that values is exact.

For 10 <= n <= 30 we have a(n) == 1 (mod 8).

Heuristically this is no coincidence. There are 8^3 = 512 tuples (x, y, z) mod 8. The frequencies of k (mod 8) for x^2 + y^3 + z^4 for k = 0 through 7 are 64, 128, 96, 32, 64, 64, 32, 32 respectively. So 1 (mod 8) has the single largest value at 128 such tuples.

Extending this to other moduli like 56 we get the largest frequencies (7168) come from 9, 17, 25 and 33 (mod 56).

The second largest frequency is 6272 which occurs at 49 (mod 56). For n = 3, 4 and 10 <= n <= 20, 22, 30 we have a(n) == 9, 17, 25, 33 or 49 (mod 56). (End)

Examples

			a(4) = 25 because 25^5 = 1852^2 + 185^3 + 8^4 = 2711^2 + 134^3 + 10^4 = 2472^2 + 150^3 + 23^4 = 2973^2 + 15^3 + 31^4 and no integer less than 25 has 4 solutions.
		

Links

• Zhining Yang, Table of n, a(n) for n = 1..70

Crossrefs

Cf. A386377, A386521.

Programs

Mathematica

f[w_]:=(v={};c=0;nn=w^5;
Do[yy=nn-z^4;Do[xx=yy-y^3;x=Sqrt@xx;
If[IntegerQ@x,If[GCD[x,y,z]==1,AppendTo[v,{x,y,z,d}];c++]],{y,Floor[yy^(1/3)]}],{z,Floor[nn^(1/4)]}];{c,w,v});
s=Table[{},20];
For[k=1,k<=100,k++,r=f[k][[1]];If[s[[r]]=={},s[[r]]=g[k];Print[s[[r]]]]]

Extensions

a(21)-a(31) from David A. Corneth, Jul 20 2025

a(32)-a(58) from Zhining Yang, Jul 31 2025