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A275168
Positive integers not of the form x^3 + 3*y^2 + z^2 with x,y,z nonnegative integers.
3
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
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}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 18 2016
STATUS
approved