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.

9, 28, 35, 54, 65, 72, 91, 126, 133, 134, 152, 182, 183, 189, 201, 217, 219, 224, 243, 250, 273, 278, 280, 309, 341, 344, 351, 370, 399, 407, 422, 432, 453, 468, 497, 513, 520, 539, 559, 576, 579, 637, 651, 658, 686, 728, 730, 737, 756, 793, 854, 855

Offset: 1

Author

Altug Alkan, Jan 19 2016

Keywords

• nonn

Comments

Inspired by intersection of A000408, A000583 and A003325.

Corresponding fourth powers are 6561, 614656, 1500625, 8503056, 17850625, 26873856, 68574961, 252047376, 312900721, 322417936, 533794816, 1097199376, 1121513121, 1275989841, 1632240801, 2217373921, 2300257521, 2517630976, 3486784401, ...

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.

Examples

			9 is a term because 9^4 = 9^3 + 18^3 = 1^2 + 28^2 + 76^2.
28 is a term because 28^4 = 28^3 + 84^3 = 64^2 + 144^2 + 768^2.
35 is a term because 35^4 = 70^3 + 105^3 = 1^2 + 600^2 + 1068^2.
54 is a term because 54^4 = 162^3 + 162^3 = 12^2 + 264^2 + 2904^2.
399 is a term because 399^4 = 665^3 + 2926^3 = 17^2 + 11236^2 + 158804^2.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000408, A000583, A003325, A267702.

Programs

PARI

isA000408(n) = {my(a, b); a=1; while(a^2+1A003325(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
for(n=3, 1e3, if(isA000408(n^4) && isA003325(n^4), print1(n, ", ")));

Extensions

Added missing term a(32), Chai Wah Wu, Jan 31 2016