A280153 - OEIS

%I #30 Jan 24 2017 17:30:25

%S 0,1,1,1,2,2,1,2,2,1,3,2,3,2,2,2,1,5,2,3,4,2,3,1,4,3,4,5,4,5,2,4,4,6,

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

%U 7,4,5,6,3,5,7,5,3,3,5,4

%N Number of ways to write n as x^3 + 2*y^3 + z^2 + 4^k, where x is a positive integer and y,z,k are nonnegative integers.

%C Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 7, 10, 17, 24, 36, 38, 52, 62, 115, 136, 185, 990.

%C (ii) Each positive integer n can be written as x^2 + 2*y^2 + z^3 + 8^k with x,y,z,k nonnegative integers.

%C (iii) Let a,b,c be positive integers with b <= c. Then any positive integer can be written as a*x^3 + b*y^2 + c*z^2 + 4^k with x,y,z,k nonnegative integers, if and only if (a,b,c) is among the following triples: (1,1,1), (1,1,2), (1,1,3), (1,1,5), (1,1,6), (1,2,3), (1,2,5), (2,1,1), (2,1,2), (2,1,3), (2,1,6), (4,1,2), (5,1,2), (8,1,2), (9,1,2).

%C We have verified that a(n) > 0 for all n = 2..10^6, and that part (ii) of the conjecture holds for all n = 1..10^6.

%C For any positive integer n, it is easy to see that at least one of n-1, n-8, n-64 is not of the form 4^k*(8m+7) with k and m nonnegative integers, thus, by the Gauss-Legendre theorem on sums of three squares, n = x^2 + y^2 + z^2 + 8^k for some nonnegative integers x,y,z and k < 3.

%C See also A280356 for a similar conjecture involving powers of two.

%H Zhi-Wei Sun, <a href="/A280153/b280153.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.

%e a(7) = 1 since 7 = 1^3 + 2*1^3 + 0^2 + 4^1.

%e a(10) = 1 since 10 = 2^3 + 2*0^3 + 1^2 + 4^0.

%e a(17) = 1 since 17 = 1^3 + 2*0^3 + 0^2 + 4^2.

%e a(24) = 1 since 24 = 2^3 + 2*0^3 + 0^2 + 4^2.

%e a(36) = 1 since 36 = 2^3 + 2*1^3 + 5^2 + 4^0.

%e a(38) = 1 since 38 = 1^3 + 2*0^3 + 6^2 + 4^0.

%e a(52) = 1 since 52 = 3^3 + 2*0^3 + 3^2 + 4^2.

%e a(62) = 1 since 62 = 2^3 + 2*1^3 + 6^2 + 4^2.

%e a(115) = 1 since 115 = 2^3 + 2*3^3 + 7^2 + 4^1.

%e a(136) = 1 since 136 = 2^3 + 2*0^3 + 8^2 + 4^3.

%e a(185) = 1 since 185 = 3^3 + 2*3^3 + 10^2 + 4^1.

%e a(990) = 1 since 990 = 7^3 + 2*3^3 + 23^2 + 4^3.

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

%t Do[r=0;Do[If[SQ[n-4^k-x^3-2y^3],r=r+1],{k,0,Log[4,n]},{x,1,(n-4^k)^(1/3)},{y,0,((n-4^k-x^3)/2)^(1/3)}];Print[n," ",r];Continue,{n,1,80}]

%Y Cf. A000290, A000302, A000578, A262813, A262827, A262857, A270533, A270559, A280356.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Dec 27 2016