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

0,3

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.

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.

Example

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].

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]

Crossrefs

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

Sequence in context: A156220 A261653 A347658 * A083900 A369030 A113517

Adjacent sequences: A343384 A343385 A343386 * A343388 A343389 A343390

Keyword

nonn

Author

Zhi-Wei Sun, Apr 13 2021

Status

approved