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

1,1

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.

Links

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

Example

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.

Prog

(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); }
T=thueinit('z^3+1);
isA003325(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, ", ")));

Crossrefs

Cf. A000408, A000583, A003325, A267702.

Sequence in context: A155473 A127629 A334185 * A024670 A141805 A256497

Adjacent sequences: A267683 A267684 A267685 * A267687 A267688 A267689

Keyword

nonn

Author

Altug Alkan, Jan 19 2016

Extensions

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

Status

approved