A275168 Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

6, 18, 23, 41, 42, 59, 78, 86, 96, 114, 115, 123, 142, 187, 195, 205, 213, 214, 240, 261, 262, 266, 303, 322, 329, 330, 383, 423, 478, 501, 510, 581, 610, 618, 642, 682, 690, 698, 761, 774, 807, 865, 870, 906, 959, 963, 990, 1206, 1222, 1230, 1234, 1302, 1312, 1314, 1320, 1346, 1411, 1697, 1706, 1781

Offset

1,1

Comments

Conjecture: The sequence has totally 150 terms as listed in the b-file the largest of which is 182842. Thus any integer n > 182842 can be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

We note that the sequence has no term greater than 182842 and not exceeding 10^6.

See also A275169 for a similar conjecture.

It is known that for any positive integers a,b,c there are infinitely many positive integers not of the form a*x^2 + b*y^2 + c*z^2 with x,y,z nonnegative integers.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..150

Example

a(1) = 6 since 1 = 0^3 + 3*0^2 + 1^2, 2 = 1^3 + 3*0^2 + 1^2, 3 = 0^3 + 3*1^2 + 0^2, 4 = 0^3 + 3*1^2 + 1^2, 5 = 1^3 + 3*1^2 + 1^2, but 6 cannot be written as x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
n=0; Do[Do[If[SQ[m-x^3-3*y^2], Goto[aa]], {x, 0, m^(1/3)}, {y, 0, Sqrt[(m-x^3)/3]}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 1800}]

Crossrefs

Cf. A000290, A000578, A022551, A022552, A262813, A270488, A274274, A275083, A275150, A275169.

Sequence in context: A373413 A358748 A081318 * A236359 A011775 A015707

Adjacent sequences: A275165 A275166 A275167 * A275169 A275170 A275171

Keyword

nonn

Author

Zhi-Wei Sun, Jul 18 2016

Status

approved