A275083 Positive integers congruent to 0 or 1 modulo 4 that cannot be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

120, 312, 813, 2136, 2680, 3224, 4404, 5340, 6420, 10060, 11320, 11824, 14008, 15856, 26544, 28804, 34392, 47984

Offset

1,1

Comments

Conjecture: (i) The sequence has totally 18 terms as listed.

(ii) For each r = 2,3 there are infinitely many positive integers n == r (mod 4) not in the form x^3 + y^2 + z^2 with x,y,z nonnegative integers.

Our computation indicates that the sequence has no other terms below 10^6.

Let d be 2 or 6. Clearly, n-d is congruent to 0 or 1 modulo 4 if n is congruent to 2 or 3 modulo 4. So part (i) of the conjecture essentially implies that for each n = 0,1,2,... either n or n-d can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers.

Links

Table of n, a(n) for n=1..18.

Example

a(1) = 120 since all those positive integers congruent to 0 or 1 modulo 4 and smaller than 120 can be written as x^3 + y^2 + z^2 with x,y,z nonnegative integers but 120 (divisible by 4) cannot be written in this way.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0; Do[If[Mod[m, 4]>1, Goto[aa]]; Do[If[SQ[m-x^3-y^2], Goto[aa]], {x, 0, m^(1/3)}, {y, 0, Sqrt[(m-x^3)/2]}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 50000}]

Crossrefs

Cf. A000290, A000578, A022551, A022552, A262813, A262857, A272979, A274274.

Sequence in context: A121898 A048190 A090391 * A098114 A135805 A327912

Adjacent sequences: A275080 A275081 A275082 * A275084 A275085 A275086

Keyword

nonn

Author

Zhi-Wei Sun, Jul 15 2016

Status

approved