A224484 Numbers which are the sum of two positive cubes and divisible by 3.

9, 54, 72, 126, 189, 243, 351, 432, 468, 513, 576, 756, 855, 945, 1008, 1125, 1332, 1395, 1458, 1512, 1674, 1755, 1944, 2205, 2322, 2331, 2457, 2709, 2745, 2808, 3087, 3402, 3456, 3528, 3591, 3744, 4104, 4221, 4608, 4914, 4941

Offset: 1

Author

Vincenzo Librandi, May 10 2013

Keywords

• nonn
• easy

Comments

If 12*h-27 is a square then some values of 3*h are in this sequence. It is easy to verify that h is of the form 3*m^2-3*m+3, and therefore 9*(m^2-m+1) = (2-m)^3+(m+1)^3.

All entries are multiples of 9. [Proof: the cubes mod 3 are A010872. So the two cubes are either of the form (3i)^3 and (3j)^3 or (3i+1)^3 and (3j+2)^3. The same 3-periodic pattern is seen in the cubes modulo 9, A167176.] - R. J. Mathar, Aug 24 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A224485 (divisible by k=5), A101421 (k=7), A101852 (k=11), A094447 (k=13), A099178 (k=17), A102619 (k=19), A101806 (k=23), A224483 (k=29), A102658 (k=31), A102618 (k=37).

Programs

Mathematica

upto[n_] := Block[{t}, Union@ Reap[ Do[If[Mod[t = x^3 + y^3, 3] == 0, Sow@ t], {x, n^(1/3)}, {y, Min[x, (n - x^3)^(1/3)]}] ][[2, 1]]]; upto[5000] (* Giovanni Resta, Jun 12 2020 *)
Module[{nn=20},Select[Union[Total/@Tuples[Range[nn]^3,2]],Mod[#,3]==0 && #Harvey P. Dale, Mar 06 2022 *)