A343391 Number of ways to write n as x^2 + [y^2/4] + [z^4/6] with x,y,z positive integers, where [.] is the floor function.

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

Offset: 1

Author

Zhi-Wei Sun, Apr 13 2021

Keywords

• nonn

Comments

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

We have verified a(n) > 0 for all n = 1..10^6.

See also A343387 for a similar conjecture.

Examples

			a(1) = 1 with 1 = 1^2 + [1^2/4] + [1^4/6].
a(2) = 1 with 2 = 1^2 + [2^2/4] + [1^4/6].
a(14) = 1 with 14 = 1^2 + [1^2/4] + [3^4/6].
a(32) = 1 with 32 = 4^2 + [8^2/4] + [1^4/6].
a(35) = 1 with 35 = 4^2 + [5^2/4] + [3^4/6].
a(77) = 1 with 77 = 8^2 + [1^2/4] + [3^4/6].
a(840) = 1 with 840 = 28^2 + [15^2/4] + [1^4/6].
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..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, A343387, A343397, A343411.

Programs

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/4]-Floor[y^4/6]],r=r+1],{x,1,Sqrt[4n+3]},{y,1,(6(n-Floor[x^2/4])+5)^(1/4)}];tab=Append[tab,r],{n,1,100}];Print[tab]