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

Author

Robert Israel, Mar 03 2020

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

Cf. A024975.

Programs

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));

Extensions

More terms from Rémy Sigrist, Mar 04 2020