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A308934 Number of ways to write n as (2^a*3^b)^2 + (2^c*3^d)^2 + x^2 + 2*y^2, where a,b,c,d,x,y are nonnegative integers with 2^a*3^b >= 2^c*3^d.

%I A308934 #8 Jul 01 2019 05:34:56
%S A308934 0,1,1,1,2,2,1,3,2,3,5,2,4,6,1,4,5,4,7,6,7,6,4,6,4,9,7,5,10,4,4,7,4,7,
%T A308934 10,7,8,9,4,8,10,7,10,9,7,11,5,6,11,7,10,8,11,11,5,14,6,9,13,3,13,9,6,
%U A308934 12,7,6,11,12,12,11,10,10,10,17,9,14,14,8,10,9,14,11,16,15,13,18,6,14,17,14,22,11,12,16,7,13,11,16,19,13
%N A308934 Number of ways to write n as (2^a*3^b)^2 + (2^c*3^d)^2 + x^2 + 2*y^2, where a,b,c,d,x,y are nonnegative integers with 2^a*3^b >= 2^c*3^d.
%C A308934 Conjecture 1: a(n) > 0 for all n > 1.
%C A308934 Conjecture 2: Any integer n > 1 can be written as (2^a*3^b)^2 + (2^c*5^d)^2 + x^2 + 2*y^2 with a,b,c,d,x,y nonnegative integers.
%C A308934 These two conjectures are similar to the Four-square Conjecture in A308734. We have verified Conjectures 1 and 2 for n up to 2*10^9 and 10^9 respectively.
%H A308934 Zhi-Wei Sun, <a href="/A308934/b308934.txt">Table of n, a(n) for n = 1..10000</a>
%e A308934 a(3) = 1 with 3 = 1^2 + 1^2 + 1^2 + 2*0^2.
%e A308934 a(7) = 1 with 7 = 2^2 + 1^2 + 0^2 + 2*1^2.
%e A308934 a(15) = 1 with 15 = 3^2 + 2^2 + 0^2 + 2*1^2.
%t A308934 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A308934 tab={};Do[r=0;Do[If[SQ[n-4^a*9^b-4^c*9^d-2x^2],r=r+1],{a,0,Log[4,n]},{b,0,Ceiling[Log[9,n/4^a]]-1},
%t A308934 {c,0,Log[4,n-4^a*9^b]},{d,0,Log[9,Min[4^(a-c)*9^b,(n-4^a*9^b)/4^c]]},{x,0,Sqrt[(n-4^a*9^b-4^c*9^d)/2]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A308934 Cf. A000079, A000244, A000290, A000351, A275344, A308734.
%K A308934 nonn
%O A308934 1,5
%A A308934 _Zhi-Wei Sun_, Jul 01 2019