

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.


3



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
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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

ZhiWei 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[nx^5Floor[y^2/7]], r=r+1], {x, 0, (n1)^(1/5)}, {y, 2, Sqrt[7(nx^5)1]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]


CROSSREFS

Cf. A000290, A000584, A270920, A338687, A343387, A343391, A343397.
Sequence in context: A348459 A266123 A115230 * A338687 A331677 A304733
Adjacent sequences: A338683 A338684 A338685 * A338687 A338688 A338689


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Apr 23 2021


STATUS

approved



