OFFSET
1,7
COMMENTS
Conjecture: a(n) > 0 for all n > 3.
This is stronger than the author's previous conjecture in A302983. It has been verified that a(n) > 0 for all n = 4..10^9.
Jiao-Min Lin (a student at Nanjing University) has found a counterexample to the conjecture: a(12558941213) = 0. - Zhi-Wei Sun, Jul 30 2022
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.
EXAMPLE
a(4) = 1 with 4 = 0^2 + 2*0^2 + 3*2^0 + 4^0.
a(5) = 1 with 5 = 1^2 + 2*0^2 + 3*2^0 + 4^0.
a(6) = 1 with 6 = 0^2 + 2*1^2 + 3*2^0 + 4^0.
a(9) = 2 with 9 = 0^2 + 2*1^2 + 3*2^0 + 4^1 = 0^2 + 2*1^2 + 3*2^1 + 4^0.
MATHEMATICA
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[MemberQ[{5, 7}, Mod[Part[Part[f[n], i], 1], 8]]&&Mod[Part[Part[f[n], i], 2], 2]==1], {i, 1, Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[QQ[n-3*2^k-4^j], Do[If[SQ[n-3*2^k-4^j-2x^2], r=r+1], {x, 0, Sqrt[(n-3*2^k-4^j)/2]}]], {k, 0, Log[2, n/3]}, {j, 0, If[3*2^k==n, -1, Log[4, n-3*2^k]]}]; tab=Append[tab, r], {n, 1, 70}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 22 2018
STATUS
approved