A332664 a(n) = number of nonnegative integers that are not the sum of {2 squares, a nonnegative 5th power, and a nonnegative n-th power}.

0, 2, 14, 115, 116, 109, 245, 381, 1387, 913, 1234, 1552, 2103, 2838, 3036, 3384, 4693, 5405, 8304, 9088, 11089, 13289, 15815, 18619, 20979, 22755, 24107, 24984, 25548

Offset: 2

Author

XU Pingya, Feb 18 2020

Keywords

• nonn
• more

Comments

a(2) = 0 by a theorem of Zhi-Wei Sun, see A273915. All terms beyond a(2) are conjectures and have only been checked to 4*10^9.

Examples

			a(2) = 0, since any nonnegative integer k is the sum of 3 squares and a nonnegative 5th power (see A273915).
a(4) = 14. Since any nonnegative integer k (<= 4*10^9) is the sum of {2 squares, a nonnegative 5th power, and a 4th power}, except for 14 numbers: 23, 44, 71, 79, 215, 383, 863, 1439, 1583, 1727, 1759, 1919, 2159, 2543.
		

Links

• W. Jagy and I. Kaplansky, Sums of Squares, Cubes and Higher Powers, Experimental Mathematics, vol. 4 (1995) pp. 169-173.

Crossrefs

Cf. A022566, A111151, A273915, A319052, A322122.

Programs

Mathematica

a(5)
Do[m=1000000 (k-1)+1; n=1000000 k;
  t=Union@Flatten@Table[x^2 + y^2 + z^5 + w^5,
{x,0,n^(1/2)}, {y,x,(n-x^2)^(1/2)}, {z,0,(n-x^2-y^2)^(1/5)},
{w, If[x^2 + y^2 + z^5 < m, Floor[(m-1-x^2-y^2-z^5)^(1/5)] + 1, z], (n-x^2-y^2-z^5)^(1/5)}];
  b=Complement[Range[m, n], t];
  Print[Length@b], {k,4000}]