cp's OEIS Frontend

A267686 Positive integers n such that n^4 = a^3 + b^3 = x^2 + y^2 + z^2 where x, y, z, a and b are positive integers, is soluble.

%I A267686 #37 Feb 05 2016 20:38:54
%S A267686 9,28,35,54,65,72,91,126,133,134,152,182,183,189,201,217,219,224,243,
%T A267686 250,273,278,280,309,341,344,351,370,399,407,422,432,453,468,497,513,
%U A267686 520,539,559,576,579,637,651,658,686,728,730,737,756,793,854,855
%N A267686 Positive integers n such that n^4 = a^3 + b^3 = x^2 + y^2 + z^2 where x, y, z, a and b are positive integers, is soluble.
%C A267686 Inspired by intersection of A000408, A000583 and A003325.
%C A267686 Corresponding fourth powers are 6561, 614656, 1500625, 8503056, 17850625, 26873856, 68574961, 252047376, 312900721, 322417936, 533794816, 1097199376, 1121513121, 1275989841, 1632240801, 2217373921, 2300257521, 2517630976, 3486784401, ...
%C A267686 2 is the first number that its 4th power, 2^4, is the sum of 2 positive cubes and is not the sum of 3 nonzero squares. 16 is the second number for this case. So 2 and 16 are not in this sequence.
%H A267686 Chai Wah Wu, <a href="/A267686/b267686.txt">Table of n, a(n) for n = 1..10000</a>
%e A267686 9 is a term because 9^4 = 9^3 + 18^3 = 1^2 + 28^2 + 76^2.
%e A267686 28 is a term because 28^4 = 28^3 + 84^3 = 64^2 + 144^2 + 768^2.
%e A267686 35 is a term because 35^4 = 70^3 + 105^3 = 1^2 + 600^2 + 1068^2.
%e A267686 54 is a term because 54^4 = 162^3 + 162^3 = 12^2 + 264^2 + 2904^2.
%e A267686 399 is a term because 399^4 = 665^3 + 2926^3 = 17^2 + 11236^2 + 158804^2.
%o A267686 (PARI) isA000408(n) = {my(a, b); a=1; while(a^2+1<n, b=1; while(b<=a && a^2+b^2<n, if(issquare(n-a^2-b^2), return(1)); b++; ); a++; ); return(0);}
%o A267686 T=thueinit('z^3+1);
%o A267686 isA003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
%o A267686 for(n=3, 1e3, if(isA000408(n^4) && isA003325(n^4), print1(n, ", ")));
%Y A267686 Cf. A000408, A000583, A003325, A267702.
%K A267686 nonn
%O A267686 1,1
%A A267686 _Altug Alkan_, Jan 19 2016
%E A267686 Added missing term a(32), _Chai Wah Wu_, Jan 31 2016