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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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 wit 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

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Last modified September 17 03:42 EDT 2021. Contains 347478 sequences. (Running on oeis4.)