A343411 Number of ways to write n as x^3 + [y^3/2] + [z^3/3] + 2^w, where [.] is the floor function, x,y,z are positive integers, and w is a nonnegative integer.

0, 1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 3, 2, 3, 4, 5, 2, 7, 2, 2, 2, 5, 4, 5, 5, 3, 3, 3, 3, 7, 6, 3, 5, 5, 6, 2, 11, 3, 6, 2, 6, 6, 8, 10, 2, 9, 2, 5, 5, 10, 5, 2, 6, 4, 4, 7, 5, 7, 2, 2, 4, 6, 7, 3, 12, 3, 7, 4, 9, 6, 5, 10, 4, 15, 4, 8, 5, 11, 4, 8, 14, 6, 4, 6, 10, 7, 8, 9, 5, 6, 4, 4, 13, 5, 7, 3, 10, 2, 7, 11

Offset: 1

Author

Zhi-Wei Sun, Apr 14 2021

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 1.

We have verified a(n) > 0 for all 1 < n <= 3*10^5.

Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021

Examples

			a(2) = 1 with 2 = 1^3 + [1^3/2] + [1^3/3] + 2^0.
a(3) = 1 with 3 = 1^3 + [1^3/2] + [1^3/3] + 2^1.
a(4) = 1 with 4 = 1^3 + [1^3/2] + [2^3/3] + 2^0.
a(6) = 1 with 6 = 1^3 + [2^3/2] + [1^3/3] + 2^0.
a(8) = 1 with 8 = 1^3 + [2^3/2] + [2^3/3] + 2^0.
a(10) = 1 with 10 = 2^3 + [1^3/2] + [1^3/3] + 2^1.
a(103) = 1 with 103 = 3^3 + [1^3/2] + [6^3/3] + 2^2.
		

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. A000079, A000578, A343326, A343368, A343384, A343387, A343391, A343397.

Programs

Mathematica

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