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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 (list; graph; refs; listen; history; text; internal format)
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?

LINKS

Table of n, a(n) for n=1..42.

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));

CROSSREFS

Cf. A336205.

Sequence in context: A090445 A228346 A269996 * A336458 A018318 A277809

Adjacent sequences:  A336237 A336238 A336239 * A336241 A336242 A336243

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

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Last modified September 30 13:46 EDT 2020. Contains 337439 sequences. (Running on oeis4.)