%I #14 Apr 24 2018 05:52:00

%S 0,1,1,2,1,2,2,2,1,3,3,5,2,3,3,2,2,5,4,5,2,3,5,2,3,5,4,7,2,4,5,3,4,6,

%T 4,7,3,6,6,4,4,5,5,9,5,6,6,2,5,5,7,8,4,5,4,4,4,6,6,8,3,6,6,3,4,6,7,8,

%U 5,8,6,5,4,6,7,8,6,6,6,2

%N Number of ways to write n as x*(3*x-2) + y*(3*y-2) + 3^u + 3^v, where x,y,u,v are integers with x <= y and 0 <= u <= v.

%C Conjecture: a(n) > 0 for all n > 1. Moreover, any integer n > 1 can be written as x*(3*x+2) + y*(3*y+2) + 3^z + 3^w, where x is an integer and y,z,w are nonnegative integers.

%C a(n) > 0 for all n = 2..3*10^8. Those x*(3*x-2) with x integral are called generalized octagonal numbers (A001082). 76683391 is the least integer n > 1 not representable as the sum of two generalized octagonal numbers and two powers of 2.

%C See also A303389, A303401 and A303432 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A303428/b303428.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.

%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.

%e a(2) = 1 with 2 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^0.

%e a(3) = 1 with 3 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^0.

%e a(4) = 2 with 4 = 1*(3*1-2) + 1*(3*1-2) + 3^0 + 3^0 = 0*(3*0-2) + 0*(3*0-2) + 3^0 + 3^1.

%e a(5) = 1 with 5 = 0*(3*0-2) + 1*(3*1-2) + 3^0 + 3^1.

%e a(9) = 1 with 9 = (-1)*(3*(-1)-2) + 0*(3*0-2) + 3^0 + 3^1.

%e a(4360) = 4 with 4360 = (-35)*(3*(-35)-2) + (-13)*(3*(-13)-2) + 3^0 + 3^4 = (-37)*(3*(-37)-2) + (-7)*(3*(-7)-2) + 3^2 + 3^2 = (-27)*(3*(-27)-2) + (-23)*(3*(-23)-2) + 3^5 + 3^5 = (-25)*(3*(-25)-2) + (-1)*(3*(-1)-2) + 3^5 + 3^7.

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

%t f[n_]:=f[n]=FactorInteger[n];

%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;

%t QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);

%t tab={};Do[r=0;Do[If[QQ[3(n-3^j-3^k)+2],Do[If[SQ[3(n-3^j-3^k-x(3x-2))+1],r=r+1],{x,-Floor[(Sqrt[3(n-3^j-3^k)/2+1]-1)/3],(Sqrt[3(n-3^j-3^k)/2+1]+1)/3}]],

%t {j,0,Log[3,n/2]},{k,j,Log[3,n-3^j]}];tab=Append[tab,r],{n,1,80}];Print[tab]

%Y Cf. A000244, A000567, A001082, A045944, A280472, A303233, A303338, A303363, A303389, A303393, A303399, A303401, A303432.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Apr 23 2018