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A273615 Numbers k such that k^4 is the average of two positive cubes while k is not.

%I A273615 #22 May 19 2023 01:36:38
%S A273615 329,518,566,662,732,741,777,804,806,876,921,998,1029,1092,1236,1238,
%T A273615 1317,1497,1526,1596,1812,1862,1929,1988,2181,2316,2604,2632,2757,
%U A273615 4204,4396,4446,4684,5068,5548,5782,5838,5856,5928,5982,6124,6126,6216
%N A273615 Numbers k such that k^4 is the average of two positive cubes while k is not.
%C A273615 If k is the average of two positive cubes, then k^4 is also the average of two positive cubes. So this sequence focuses on the solutions that are not trivial.
%H A273615 Robert Israel, <a href="/A273615/b273615.txt">Table of n, a(n) for n = 1..502</a>
%e A273615 329 is a term because 329 is not the average of two positive cubes while 329^4 = (1833^3 + 2585^3)/2.
%p A273615 Q:=  proc(x) local t;
%p A273615   for t in select(t -> t^3<=x and 4*t^3 > x and x/t - t^2 mod 3 = 0,
%p A273615         numtheory:-divisors(x)) do
%p A273615     if issqr((x/t - t^2)/3)  then return true fi
%p A273615   od:
%p A273615   false
%p A273615 end proc:
%p A273615 select(x -> not(Q(x)) and Q(x^4), [$1..10000]); # _Robert Israel_, May 26 2016
%t A273615 Q[x_] := Module[{s, t}, s = Select[Divisors[x], #^3 <= x && 4*#^3 > x && Mod[x/# - #^2, 3] == 0 &]; For[t = 1, t <= Length[s], t++, If[IntegerQ@Sqrt[(x/s[[t]] - s[[t]]^2)/3],  Return[True]]]; False];
%t A273615 Reap[For[x = 1, x <= 10000, x++, If[!Q[x] && Q[x^4], Print[x]; Sow[x]]]][[2, 1]] (* _Jean-François Alcover_, May 18 2023, after _Robert Israel_ *)
%o A273615 (PARI) isA003325(n) = for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1));
%o A273615 lista(nn) = for(n=1, nn, if(isA003325(2*n^4) && !isA003325(2*n), print1(n, ", ")));
%o A273615 (PARI) T=thueinit('z^3+1);
%o A273615 isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
%o A273615 is(n)=isA003325(2*n^4) && !isA003325(2*n) \\ _Charles R Greathouse IV_, May 27 2016
%Y A273615 Cf. A003325, A051387.
%K A273615 nonn
%O A273615 1,1
%A A273615 _Altug Alkan_, May 26 2016