A327586 Numbers k such that k^4 = a^3 + b^3 + c^3 for some pairwise coprime positive integers a,b,c.

39, 57, 70, 74, 106, 111, 147, 174, 209, 216, 236, 237, 244, 252, 291, 300, 318, 327, 333, 336, 342, 360, 366, 372, 387, 403, 417, 424, 450, 462, 489, 524, 540, 561, 582, 594, 615, 624, 636, 638, 651, 660, 673, 696, 700, 714, 739, 741, 768, 771, 804, 827, 837

Offset

1,1

Comments

a(10) = 216 is the least term whose fourth power has two representations as a sum of the cubes of three pairwise coprime positive integers: 216^4 = 1217^3 + 639^3 + 484^3 = 1257^3 + 575^3 + 82^3. - Rémy Sigrist, Mar 04 2020

The least terms with 3 and 4 representations are a(230)=4914 and a(269)=5832, respectively. - Giovanni Resta, Mar 04 2020

Links

Giovanni Resta, Table of n, a(n) for n = 1..549 (terms < 12000)

Mathematics StackExchange, Sum of three perfect cubes is equal to a perfect fourth

Giovanni Resta, Representations of k^4 for k<12000

Example

a(3) = 70 is a term because 70^4 = 81^3 + 167^3 + 266^3, and 81, 167 and 266 are positive and pairwise coprime.

Maple

N:= 200: # to get all terms <= N
qmax:= N^4: Res:= {}:
for a from 1 while a^3 < qmax do
  for b from a+1 while a^3 + b^3 < qmax do
    if igcd(a, b) <> 1 then next fi;
    for c from b+1 while a^3 + b^3 + c^3 <= qmax do
      if igcd(c, a*b) <> 1 then next fi;
        q:= a^3 + b^3 + c^3;
        if issqr(q) and issqr(sqrt(q)) then
        Res:= Res union  {sqrt(sqrt(q))};
      fi
od od od:
sort(convert(Res, list));

Crossrefs

Cf. A024975.

Sequence in context: A063480 A305026 A009633 * A156333 A116281 A168530

Adjacent sequences: A327583 A327584 A327585 * A327587 A327588 A327589

Keyword

nonn

Author

Robert Israel, Mar 03 2020

Extensions

More terms from Rémy Sigrist, Mar 04 2020

Status

approved