A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

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

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..7*10^6.

See also A338687 for a similar conjecture.

Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021

Examples

			a(1) = 1 with 1 = 0^5 + 1^2 + [2^2/7].
a(166) = 1 with 166 = 0^5 + 1^2 + [34^2/7].
a(323) = 1 with 323 = 2^5 + 17^2 + [4^2/7].
a(815) = 1 with 815 = 2^5 + 1^2 + [74^2/7].
a(2069) = 1 with 2069 = 0^5 + 37^2 + [70^2/7].
a(7560) = 1 with 7560 = 2^5 + 64^2 + [155^2/7].
a(24195) = 1 with 24195 = 0^5 + 8^2 + [411^2/7].
a(90886) = 2 with 90886 = 4^5 + 34^2 + [788^2/7] = 9^5 + 139^2 + [296^2/7].
		

Links

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

Crossrefs

Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[n-x^5-Floor[y^2/7]],r=r+1],{x,0,(n-1)^(1/5)},{y,2,Sqrt[7(n-x^5)-1]}];tab=Append[tab,r],{n,1,100}];Print[tab]