login
Number of ways to write n as x^2 + [y^2/2] + [z^4/8], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.
8

%I #17 Apr 15 2021 04:45:07

%S 1,1,2,2,3,2,3,1,3,2,3,4,3,4,4,3,3,1,6,4,3,3,4,3,3,2,4,5,4,4,3,2,3,4,

%T 5,6,5,4,6,2,6,4,4,7,5,3,4,1,5,4,8,8,2,5,5,1,5,4,3,8,5,6,2,3,5,4,6,4,

%U 6,4,5,3,5,4,4,5,8,2,7,2,3,7,6,9,3,6,10,5,5,5,5,8,3,5,3,6,7,3,9,8,6

%N Number of ways to write n as x^2 + [y^2/2] + [z^4/8], where [.] is the floor function, x is a nonnegative integer, and y and z are positive integers.

%C Conjecture: a(n) > 0 for all n >= 0.

%C This has been verified for all n = 0..10^5.

%C We also conjecture that each n = 0,1,... can be written as x^2 + [y^2/3] + [z^4/7] with x,y,z nonnegative integers.

%C See also A343391 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A343387/b343387.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1504.01608">Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c]</a>, arXiv:1504.01608 [math.NT], 2015.

%e a(0) = 1 with 0 = 0^2 + [1^2/2] + [1^4/8].

%e a(47) = 1 with 47 = 5^2 + [5^2/2] + [3^4/8].

%e a(55) = 1 with 55 = 7^2 + [3^2/2] + [2^4/8].

%e a(217) = 1 with 217 = 11^2 + [6^2/2] + [5^4/8].

%e a(377) = 1 with 377 9^2 + [23^2/2] + [4^4/8].

%e a(392) = 1 with 392 = 0^2 + [28^2/2] + [1^4/8].

%e a(734) = 1 with 734 = 12^2 + [32^2/2] + [5^4/8].

%e a(1052) = 1 with 1052 = 32^2 + [6^2/2] + [3^4/8].

%e a(1054) = 1 with 1054 = 30^2 + [17^2/2] + [3^4/8].

%e a(1817) = 1 with 1817 = 39^2 + [23^2/2] + [4^4/8].

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[n-Floor[x^2/2]-Floor[y^4/8]],r=r+1],{x,1,Sqrt[2n+1]},{y,1,(8(n-Floor[x^2/2])+7)^(1/4)}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A343326, A343368, A343384, A343391, A343397, A343411.

%K nonn

%O 0,3

%A _Zhi-Wei Sun_, Apr 13 2021