A343397 Number of ways to write n as 2^x + [y^2/3] + [z^2/4] with x,y,z positive integers, where [.] is the floor function.

0, 1, 2, 3, 4, 4, 5, 5, 8, 5, 9, 5, 8, 8, 6, 9, 9, 10, 8, 11, 10, 10, 9, 9, 14, 8, 8, 10, 12, 11, 6, 14, 13, 10, 12, 13, 15, 11, 13, 9, 20, 6, 12, 17, 13, 13, 10, 11, 17, 12, 11, 13, 15, 14, 9, 13, 13, 14, 11, 18, 11, 15, 7, 12, 22, 13, 14, 17, 17, 11, 15, 13, 24, 16, 9, 17, 15, 15, 14, 18

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 1.

We have verified a(n) > 0 for all n = 2..2*10^6.

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 = 2^1 + [1^2/3] + [1^2/4].
a(3) = 2 with 3 = 2^1 + [1^2/3] + [2^2/4] = 2^1 + [2^2/3] + [1^2/4].

Mathematica

PowQ[n_]:=PowQ[n]=n>1&&IntegerQ[Log[2, n]];
tab={}; Do[r=0; Do[If[PowQ[n-Floor[x^2/3]-Floor[y^2/4]], r=r+1], {x, 1, Sqrt[3n-1]}, {y, 1, Sqrt[4(n-Floor[x^2/3]-1)+1]}]; tab=Append[tab, r], {n, 1, 80}]; Print[tab]

Crossrefs

Cf. A000079, A000290, A343326, A343368, A343384, A343387, A343391, A343411.

Sequence in context: A216411 A110532 A049987 * A270832 A257646 A051898

Adjacent sequences: A343394 A343395 A343396 * A343398 A343399 A343400

Keyword

nonn

Author

Zhi-Wei Sun, Apr 13 2021

Status

approved