A352629 - OEIS

%I #13 Mar 26 2022 21:19:39

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

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

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

%N Number of ways to write n as a^2 + 2*b^2 + c^4 + 9*c^4 + 3*c^2*d^2, where a,b,c,d are nonnegative integers.

%C Conjecture: a(n) > 0 for all n = 0,1,2,.... In other words, any nonnegative integer can be written as a^2 + 2*b^2 + c^4 + 9*d^4 + 3*c^2*d^2 with a,b,c,d integers.

%C It seems that a(n) = 1 only for n = 0, 5, 6, 7, 8, 14, 23, 29, 30, 71, 74, 78, 93, 143, 197, 221, 266, 312, 407, 453, 586, 703, 716, 794.

%C See also A352627, A352628 and A352632 for similar conjectures.

%H Zhi-Wei Sun, <a href="/A352629/b352629.txt">Table of n, a(n) for n = 0..10000</a>

%e a(30) = 1 with 30 = 3^2 + 2*2^2 + 1^4 + 9*1^4 + 3*1^2*1^2.

%e a(71) = 1 with 71 = 4^2 + 2*3^2 + 2^4 + 9*1^4 + 3*2^2*1^2.

%e a(312) = 1 with 312 = 15^2 + 2*5^2 + 2^4 + 9*1^4  + 3*2^2*1^2.

%e a(703) = 1 with 703 = 26^2 + 2*3^2 + 0^4 + 9*1^4 + 3*0^2*1^2.

%e a(716) = 1 with 716 = 18^2 + 2*14^2 + 0^4 + 9*0^4 + 3*0^2*0^2.

%e a(794) = 1 with 794 = 13^2 + 2*0^2 + 5^4 + 9*0^4 + 3*5^2*0^2.

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

%t tab={};Do[r=0;Do[If[SQ[n-9d^4-c^4-3c^2*d^2-2b^2],r=r+1],{d,0,(n/9)^(1/4)},{c,0,Sqrt[(Sqrt[4n-27*d^4]-3d^2)/2]},{b,0,Sqrt[(n-9d^4-c^4-3c^2*d^2)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000290, A000583, A352627, A352628, A352632.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 24 2022