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

1,5

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

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(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.

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]

Crossrefs

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

Sequence in context: A257523 A067044 A325677 * A287477 A231473 A216652

Adjacent sequences: A343408 A343409 A343410 * A343412 A343413 A343414

Keyword

nonn

Author

Zhi-Wei Sun, Apr 14 2021

Status

approved