A343387 Number of ways to write n as x^2 + [y^2/2] + [z^4/8], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

1, 1, 2, 2, 3, 2, 3, 1, 3, 2, 3, 4, 3, 4, 4, 3, 3, 1, 6, 4, 3, 3, 4, 3, 3, 2, 4, 5, 4, 4, 3, 2, 3, 4, 5, 6, 5, 4, 6, 2, 6, 4, 4, 7, 5, 3, 4, 1, 5, 4, 8, 8, 2, 5, 5, 1, 5, 4, 3, 8, 5, 6, 2, 3, 5, 4, 6, 4, 6, 4, 5, 3, 5, 4, 4, 5, 8, 2, 7, 2, 3, 7, 6, 9, 3, 6, 10, 5, 5, 5, 5, 8, 3, 5, 3, 6, 7, 3, 9, 8, 6

Offset: 0

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n >= 0.

This has been verified for all n = 0..10^5.

We also conjecture that each n = 0,1,... can be written as x^2 + [y^2/3] + [z^4/7] with x,y,z nonnegative integers.

See also A343391 for a similar conjecture.

Examples

			a(0) = 1 with 0 = 0^2 + [1^2/2] + [1^4/8].
a(47) = 1 with 47 = 5^2 + [5^2/2] + [3^4/8].
a(55) = 1 with 55 = 7^2 + [3^2/2] + [2^4/8].
a(217) = 1 with 217 = 11^2 + [6^2/2] + [5^4/8].
a(377) = 1 with 377 9^2 + [23^2/2] + [4^4/8].
a(392) = 1 with 392 = 0^2 + [28^2/2] + [1^4/8].
a(734) = 1 with 734 = 12^2 + [32^2/2] + [5^4/8].
a(1052) = 1 with 1052 = 32^2 + [6^2/2] + [3^4/8].
a(1054) = 1 with 1054 = 30^2 + [17^2/2] + [3^4/8].
a(1817) = 1 with 1817 = 39^2 + [23^2/2] + [4^4/8].
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.

Crossrefs

Cf. A000290, A000583, A343326, A343368, A343384, A343391, A343397, A343411.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/2]-Floor[y^4/8]],r=r+1],{x,1,Sqrt[2n+1]},{y,1,(8(n-Floor[x^2/2])+7)^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]