OFFSET
1,5
COMMENTS
Conjecture 1: a(n) > 0 for all n > 1.
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.
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.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(3) = 1 with 3 = 1^2 + 1^2 + 1^2 + 2*0^2.
a(7) = 1 with 7 = 2^2 + 1^2 + 0^2 + 2*1^2.
a(15) = 1 with 15 = 3^2 + 2^2 + 0^2 + 2*1^2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
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},
{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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 01 2019
STATUS
approved