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A336240
Numbers k such that k = x^2+y^2+z^2 = x^3+y^3+z^3 for some integers x,y,z.
1
0, 1, 2, 3, 6, 27, 29, 354, 729, 2027, 6859, 7778, 19846, 20577, 23277, 35937, 58754, 130979, 132651, 232282, 265602, 332750, 389017, 499853, 885602, 970299, 1492779, 2146689, 2413154, 3764477, 4330747, 5694978, 5929741, 8120601, 8388227, 12068354, 14348907, 17005629, 23522402, 24137569, 31999403, 34328125
OFFSET
1,3
COMMENTS
Contains (2*m^2 + 1)^3 corresponding to x=2*m^2+1, y=-z=2*m^3+m, and m^6/2 - 3*m^2/2 + 3 corresponding to x=-m^2+1, y=-m^3/2+m/2+1, z=m^3/2-m/2+1.
Are there other infinite parametric families of solutions?
EXAMPLE
a(6)=27 is in the sequence because 27 = (-3)^2 + 3^2 + 3^2 = (-3)^3 + 3^3 + 3^3.
a(7)=29 is in the sequence because 29 = (-2)^2 + (-3)^2 + 4^2 = (-2)^3 + (-3)^3 + 4^3.
MAPLE
N:= 2*10^5: # for all terms <= N
R:= NULL:
for xx from 0 while 3*xx^2 <= N do
for yy from xx while xx^2 + 2*yy^2 <= N do
for zz from yy while xx^2 + yy^2 + zz^2 <= N do
t:= xx^2 + yy^2 + zz^2;
c:= [xx^3, yy^3, zz^3];
if member(t, {seq(seq(seq(e1*c[1]+e2*c[2]+e3*c[3], e1=[-1, 1]), e2=[-1, 1]), e3=[-1, 1])}) then R:= R, t; fi
od od od:
sort(convert({R}, list));
MATHEMATICA
NN = 2*10^5; (* for all terms <= NN *)
R = {};
Module[{x, y, z, t, c}, For[x = 0, 3*x^2 <= NN, x++, For[y = x, x^2 + 2^2 <= NN, y++, For[z = y, x^2 + y^2 + z^2 <= NN, z++, t = x^2 + y^2 + z^2; c = {x^3, y^3, z^3}; If[MemberQ[Flatten@Table[{e1, e2, e3}. c, {e1, {-1, 1}}, {e2, {-1, 1}}, {e3, {-1, 1}}], t], Print[t]; AppendTo[R, t]]]]]];
R // Union (* Jean-François Alcover, Aug 11 2023, after Robert Israel *)
CROSSREFS
Cf. A336205.
Sequence in context: A090445 A228346 A269996 * A336458 A348867 A018318
KEYWORD
nonn
AUTHOR
Robert Israel, Jul 13 2020
EXTENSIONS
a(27)-a(35) from David A. Corneth, Jul 13 2020
a(36)-a(42) from Andrew R. Booker, Jul 14 2020
STATUS
approved