A338686 Number of ways to write n as x^5 + y^2 + [z^2/7], where x,y,z are integers with x >= 0, y >= 1 and z >= 2, and [.] is the floor function.

1, 2, 2, 3, 3, 3, 3, 2, 3, 4, 4, 4, 3, 2, 3, 4, 3, 5, 5, 2, 4, 3, 2, 3, 4, 5, 4, 4, 4, 4, 2, 2, 6, 5, 2, 6, 8, 6, 4, 5, 6, 6, 5, 4, 6, 5, 4, 5, 4, 10, 6, 5, 8, 3, 5, 5, 7, 6, 4, 5, 7, 5, 2, 6, 7, 6, 7, 8, 6, 4, 5, 6, 8, 6, 2, 4, 8, 4, 6, 3, 5, 10, 6, 8, 7, 5, 5, 6, 5, 5, 5, 7, 6, 4, 4, 6, 3, 8, 7, 7

Offset

1,2

Comments

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

We have verified a(n) > 0 for all n = 1..7*10^6.

See also A338687 for a similar conjecture.

Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 with 1 = 0^5 + 1^2 + [2^2/7].
a(166) = 1 with 166 = 0^5 + 1^2 + [34^2/7].
a(323) = 1 with 323 = 2^5 + 17^2 + [4^2/7].
a(815) = 1 with 815 = 2^5 + 1^2 + [74^2/7].
a(2069) = 1 with 2069 = 0^5 + 37^2 + [70^2/7].
a(7560) = 1 with 7560 = 2^5 + 64^2 + [155^2/7].
a(24195) = 1 with 24195 = 0^5 + 8^2 + [411^2/7].
a(90886) = 2 with 90886 = 4^5 + 34^2 + [788^2/7] = 9^5 + 139^2 + [296^2/7].

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-x^5-Floor[y^2/7]], r=r+1], {x, 0, (n-1)^(1/5)}, {y, 2, Sqrt[7(n-x^5)-1]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.

Sequence in context: A266123 A115230 A363937 * A338687 A331677 A304733

Adjacent sequences: A338683 A338684 A338685 * A338687 A338688 A338689

Keyword

nonn

Author

Zhi-Wei Sun, Apr 23 2021

Status

approved