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

%I A327586 #30 Mar 04 2020 09:42:41
%S A327586 39,57,70,74,106,111,147,174,209,216,236,237,244,252,291,300,318,327,
%T A327586 333,336,342,360,366,372,387,403,417,424,450,462,489,524,540,561,582,
%U A327586 594,615,624,636,638,651,660,673,696,700,714,739,741,768,771,804,827,837
%N A327586 Numbers k such that k^4 = a^3 + b^3 + c^3 for some pairwise coprime positive integers a,b,c.
%C A327586 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
%C A327586 The least terms with 3 and 4 representations are a(230)=4914 and a(269)=5832, respectively. - _Giovanni Resta_, Mar 04 2020
%H A327586 Giovanni Resta, <a href="/A327586/b327586.txt">Table of n, a(n) for n = 1..549</a> (terms < 12000)
%H A327586 Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/3568046/sum-of-three-perfect-cubes-is-equal-to-a-perfect-fourth/3568103">Sum of three perfect cubes is equal to a perfect fourth</a>
%H A327586 Giovanni Resta, <a href="/A327586/a327586.txt">Representations of k^4 for k<12000</a>
%e A327586 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.
%p A327586 N:= 200: # to get all terms <= N
%p A327586 qmax:= N^4: Res:= {}:
%p A327586 for a from 1 while a^3 < qmax do
%p A327586   for b from a+1 while a^3 + b^3 < qmax do
%p A327586     if igcd(a,b) <> 1 then next fi;
%p A327586     for c from b+1 while a^3 + b^3 + c^3 <= qmax do
%p A327586       if igcd(c,a*b) <> 1 then next fi;
%p A327586         q:= a^3 + b^3 + c^3;
%p A327586         if issqr(q) and issqr(sqrt(q)) then
%p A327586         Res:= Res union  {sqrt(sqrt(q))};
%p A327586       fi
%p A327586 od od od:
%p A327586 sort(convert(Res,list));
%Y A327586 Cf. A024975.
%K A327586 nonn
%O A327586 1,1
%A A327586 _Robert Israel_, Mar 03 2020
%E A327586 More terms from _Rémy Sigrist_, Mar 04 2020