OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 11, 17, 18, 47, 108, 109, 234, 359. Also, any nonnegative integer can be written as x^6 + y^3 + z*(z+1) + w*(w+1), where x,y,z,w are nonnegative integers with x <= 2.
We have verified a(n) > 0 for all n = 0..2*10^7.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
EXAMPLE
a(11) = 1 with 11 = 1^9 + 2^3 + 0*1 + 1*2.
a(18) = 1 with 18 = 0^9 + 0^3 + 2*3 + 3*4.
a(109) = 1 with 109 = 1^9 + 4^3 + 1*2 + 6*7.
a(234) = 1 with 234 = 0^9 + 6^3 + 2*3 + 3*4.
a(359) = 1 with 359 = 1^9 + 2^3 + 10*11 + 15*16.
a(1978) = 3 with 1978 = 2^9 + 2^3 + 26*27 + 27*28 = 2^9 + 6^3 + 19*20 + 29*30 = 2^9 + 6^3 + 24*25 + 25*26.
MATHEMATICA
TQ[n_]:=TQ[n]=IntegerQ[Sqrt[4n+1]];
tab={}; Do[r=0; Do[If[TQ[n-x^9-y^3-z(z+1)], r=r+1], {x, 0, Min[2, n^(1/9)]}, {y, 0, (n-x^9)^(1/3)}, {z, 0, (Sqrt[2(n-x^9-y^3)+1]-1)/2}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jan 31 2019
STATUS
approved