A267702 Numbers that are the sum of 3 nonzero squares (A000408) and the sum of 2 positive cubes (A003325).

9, 35, 54, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 250, 280, 341, 344, 370, 432, 468, 513, 539, 576, 637, 686, 728, 730, 737, 756, 793, 854, 945, 1001, 1027, 1064, 1072, 1125, 1216, 1241, 1332, 1339, 1358, 1395, 1456, 1458, 1512, 1547, 1674, 1729, 1736, 1755, 1843, 1853

Offset

1,1

Comments

Intersection of A000408 and A003325.

Sequence focuses on the solutions of equation x^3 + y^3 = a^2 + b^2 + c^2 where x, y, a, b, c > 0.

Links

Robert Israel, Table of n, a(n) for n = 1..3649

Example

9 is a term because 9 = 1^3 + 2^3 = 1^2 + 2^2 + 2^2.
35 is a term because 35 = 2^3 + 3^3 = 1^2 + 3^2 + 5^2.
54 is a term because 54 = 3^3 + 3^3 = 3^2 + 3^2 + 6^2.

Maple

N:= 1000: # to get all terms <= N
S3:= {seq(seq(seq(a^2+b^2+c^2, c = b .. floor(sqrt(N-a^2-b^2))),
b=a .. floor(sqrt((N-a^2)/2))), a = 1 .. floor(sqrt(N/3)))}:
C2:= {seq(seq(a^3+b^3, b = a .. floor((N-a^3)^(1/3))), a = 1 .. floor((N/2)^(1/3)))}:
sort(convert(S3 intersect C2, list)); # Robert Israel, Jan 25 2016

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, 1e4, if(isA000408(n) && isA003325(n), print1(n, ", ")));

Crossrefs

Cf. A000408, A003325.

Sequence in context: A187554 A379128 A338010 * A339995 A085366 A304913

Adjacent sequences: A267699 A267700 A267701 * A267703 A267704 A267705

Keyword

nonn

Author

Altug Alkan, Jan 23 2016

Status

approved