A337366 - OEIS

%I #14 Sep 11 2020 10:56:58

%S 1,0,1,1,2,1,1,2,1,4,6,3,8,8,14,7

%N Number of representations of A036691(n) as a sum of 3 nonnegative cubes.

%C Conjecture I: a(n) = 0 only for n = 1. That is, any product of first n > 1 composite numbers is a sum of at most 3 positive cubes. For example,

%C A036691(100) = 2563573191821442299652988946477367093137353211904000000000^3 + 21431289850849406740917647451954098598503667204096000000000^3 + 26409890400237152457638095665189553529771293409280000000000^3.

%C Conjecture II: For any term t >= 1, there are only finitely many values of n such that a(n) = t.

%F a(n) = A025447(A036691(n)).

%e a(4) = 2 because A036691(4) = 1728 = 12^3 = 6^3 + 8^3 + 10^3.

%t A036691 = Join[{1}, FoldList[Times, Select[Range[20], CompositeQ]]];

%t Table[Length@ PowersRepresentations[A036691[[n]], 3, 3], {n, 10}] (* _Robert Price_, Sep 08 2020 *)

%Y Cf. A025447, A036691, A226955, A267414.

%K nonn,more

%O 0,5

%A _Altug Alkan_, Aug 25 2020