A202173 - OEIS

%I #65 Oct 13 2015 13:35:32

%S 76,148,183,356,418,428,445,491,580,671,788,931,967,1084,1121,1184,

%T 1210,1219,1228,1247,1499,1508,1562,1618,1723,1975,2020,2129,2164,

%U 2236,2300,2332,2362,2372,2452,2470,2561,2722,2794,2857

%N Conjectured list of numbers not of the form x^3 + y^3 + 2*z^3.

%C Wacław Sierpiński asked if a(1) = 76.

%C Searched -50000 <= x,y <= 50000 for terms through 3000.

%C Seiji Tomita, on the math-fun mailing list (May 18 2012), gives

%C      1444 = -24062122787^3 - 9841546529^3 + 2*19524116332^3

%C      1462 = 111091225^3 - 110862443^3 - 2*16168112^3

%C      1588 = -6314285^3 - 6232583^3 + 2*6273700^3

%C      2246 = -7194061^3 - 2344975^3 + 2*5775101^3

%C      2822 = 8070731^3 - 3630235^3 - 2*6205213^3

%C Very little is known about this sequence; I do not know if any of these terms are correct. Can the sequence be proved nonempty? Theorem 3.3 in Broughan shows that modular arguments will not suffice. - _Charles R Greathouse IV_, Jun 12 2012

%C Hypothesis: the sequence is empty. See the Cohen book for details. - _Arkadiusz Wesolowski_, Aug 20 2013

%D Henri Cohen, Number Theory. Volume I: Tools and Diophantine Equations, Graduate Texts in Mathematics 239, Springer, 2007, p. 381.

%D Chao Ko: Decompositions into four cubes. Journ. London Math. Soc., 11 (1936), pp. 218-219.

%D L. J. Mordell: On the four integer cubes problem. Journ. London Math. Soc., 11 (1936), pp. 208-218.

%D Wacław Sierpiński, O stu prostych, ale trudnych zagadnieniach arytmetyki. Warsaw: PZWS, 1959, pp. 26, 72.

%H Kevin A. Broughan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Broughan/broughan25.html">Characterizing the sum of two cubes</a>, J. Integer Seqs., Vol. 6, 2003.

%H Eckford Cohen, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa5/aa544.pdf">Arithmetical functions associated with arbitrary sets of integers</a>, Acta Arithmetica 5 (1959), pp. 407-415.

%H Andrzej Makowski, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa5/aa519.pdf">Sur quelques problèmes concernant les sommes de quatre cubes</a>, Acta Arithmetica 5 (1959), pp. 121-123.

%H A. Schinzel and W. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa414.pdf">Sur les sommes de quatre cubes</a>, Acta Arithmetica 4 (1958), pp. 20-30.

%e 37 can be expressed as 2^3 + 3^3 + 2*1^3 or (-12)^3 + 13^3 + 2*(-6)^3, so 37 is not in the sequence.

%t Complement[Range[98], Flatten[Table[x^3 + y^3 + 2*z^3, {x, -35, 143}, {y, -62, 259}, {z, -209, 52}]]]

%Y Cf. A014136.

%K nonn

%O 1,1

%A _Charles R Greathouse IV_ and _Arkadiusz Wesolowski_, Dec 22 2011