A338687 Number of ways to write n as x^4 + y^2 + floor(z^2/7), where x,y,z are integers with x >= 0, y >= 1 and z >= 2.

1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 4, 3, 2, 3, 4, 4, 6, 6, 4, 5, 5, 3, 4, 6, 7, 6, 6, 5, 5, 4, 4, 6, 8, 2, 5, 10, 4, 5, 5, 7, 6, 5, 4, 7, 6, 2, 5, 6, 7, 5, 8, 8, 4, 5, 6, 6, 6, 3, 4, 9, 3, 4, 5, 6, 9, 8, 7, 5, 4, 5, 6, 8, 6, 1, 6, 6, 5, 5, 5, 4, 11, 9, 7, 9, 6, 7, 7, 8, 5, 8, 8, 8, 6, 6, 5, 7, 8, 10, 10

Offset: 1

Author

Zhi-Wei Sun, Apr 23 2021

Keywords

• nonn

Comments

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

We have verified a(n) > 0 for all n = 1..5*10^6.

See also A338686 for a similar conjecture.

Examples

			a(1) = 1 with 1 = 0^4 + 1^2 + floor(2^2/7).
a(75) = 1 with 75 = 0^4 + 8^2 + floor(9^2/7).
a(1799) = 1 with 1799 = 5^4 + 25^2 + floor(62^2/7).
a(7224) = 1 with 7224 = 9^4 + 19^2 + floor(46^2/7).
a(27455) = 2 with 27455 = 0^4 + 7^2 + floor(438^2/7) = 8^4 + 118^2 + floor(257^2/7).
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000290, A000583, A270566, A338686, A338696, A343387, A343391, A343397.

Programs

Mathematica

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