OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n = 0,1,2,....
This has been verified for all n = 0..10^8.
It seems that a(n) = 1 only for n = 0, 41, 131, 141, 145, 225, 251, 297, 591, 621, 916, 1021, 1241, 1431, 2025, 4691.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120.
EXAMPLE
a(145) = 1 with 145 = 0^8 + 3^4 + 2*0^4 + 4*2^4 + 0*1.
a(225) = 1 with 225 = 1^8 + 2^4 + 2*3^4 + 4*1^4 + 6*7.
a(916) = 1 with 916 = 2^8 + 2^4 + 2*4^4 + 4*0^4 + 11*12.
a(1021) = 1 with 1021 = 0^8 + 5^4 + 2*0^4 + 4*3^4 + 8*9.
a(1241) = 1 with 1241 = 0^8 + 5^4 + 2*0^4 + 4*2^4 + 23*24.
a(1431) = 1 with 1431 = 1^8 + 6^4 + 2*1^4 + 4*0^4 + 11*12.
a(2025) = 1 with 2025 = 2^8 + 3^4 + 2*2^4 + 4*3^4 + 36*37.
a(4691) = 1 with 4691 = 2^8 + 3^4 + 2*0^4 + 4*2^4 + 65*66.
MATHEMATICA
QQ[n_]:=QQ[n]=IntegerQ[Sqrt[4n+1]];
tab={}; Do[r=0; Do[If[QQ[n-w^8-4z^4-2y^4-x^4], r=r+1], {w, 0, n^(1/8)}, {z, 0, ((n-w^8)/4)^(1/4)}, {y, 0, ((n-w^8-4z^4)/2)^(1/4)}, {x, 0, (n-w^8-4z^4-2y^4)^(1/4)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 26 2022
STATUS
approved