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

1,3

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.

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.

Example

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].

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]

Crossrefs

Cf. A000290, A000583, A343326, A343368, A343384, A343387, A343397, A343411.

Sequence in context: A295784 A275803 A060131 * A024677 A276856 A174296

Adjacent sequences: A343388 A343389 A343390 * A343392 A343393 A343394

Keyword

nonn

Author

Zhi-Wei Sun, Apr 13 2021

Status

approved