cp's OEIS Frontend

A261029 Number of ways to write n in the form F(x,y,z) = x^3 + y^3 + z^3 - 3xyz, where 0 <= x <= y <= z and z >= x+1.

%I A261029 #83 Mar 04 2025 02:46:56
%S A261029 0,1,1,0,1,1,0,1,2,1,1,1,0,1,1,0,2,1,1,1,2,0,1,1,0,1,1,2,3,1,0,1,2,0,
%T A261029 1,2,1,1,1,0,2,1,0,1,2,1,1,1,0,2,1,0,2,1,3,1,3,0,1,1,0,1,1,1,3,2,0,1,
%U A261029 2,0,2,1,2,1,1,0,2,2,0,1,2,3,1,1,0,1,1
%N A261029 Number of ways to write n in the form F(x,y,z) = x^3 + y^3 + z^3 - 3xyz, where 0 <= x <= y <= z and z >= x+1.
%C A261029 The following is a short proof of the corresponding 1915 result of R. D. Carmichael for a weaker restriction.
%C A261029 If n is in A074232, then a(n) >= 1, in view of the following identities: if n == 1 (mod 3), then n = F((n-1)/3, (n-1)/3, (n+2)/3); if n == 2 (mod 3), then n = F((n-2)/3, (n+1)/3, (n+1)/3); if n == 0 (mod 9), then n = F(n/9-1, n/9, n/9+1). QED
%C A261029 Further, if n > 1 is the cube of a positive number or the sum of two positive cubes, except for 2 and 9, then a(n) >= 2.
%C A261029 The sequence is unbounded.
%C A261029 Proof. We use the homogeneity of F(x,y,z) of degree 3. By induction, show that a(8^k) >= k+1. It is evident for k=0. Suppose that it is true for some value of k. Take k+1 triples (x_i,y_i,z_i) such that 8^k = F(x_i, y_i, z_i), i=1,...,k+1. Then for k+1 triples of even numbers (2*x_i, 2*y_i, 2*z_i) we have 8^(k+1) = F(2*x_i, 2*y_i, 2*z_i). But there is always a triple of not all even numbers (x=(n-1)/3, y=(n-1)/3, z=(n+2)/3) or (x=(n-2)/3, y=(n+1)/3, z=(n+1)/3), where n = 8^(k+1), for which 8^(k+1) = F(x,y,z). So a(8^(k+1)) >= k+2. QED
%C A261029 Theorem. For every n there exists k such that a(k)=n. For a proof, see [Shevelev] link.
%C A261029 Smallest such k are presented in sequence A260935.
%H A261029 Peter J. C. Moses and Chai Wah Wu, <a href="/A261029/b261029.txt">Table of n, a(n) for n = 0..10000</a> (terms for n = 0..999 from Peter J. C. Moses)
%H A261029 R. D. Carmichael, <a href="http://dx.doi.org/10.1090/S0002-9904-1915-02730-8">On the representation of numbers in the form x^3+y^3+z^3-3xyz</a>, Bull. Amer. Math. Soc. 22 (1915), 111-117.
%H A261029 Peter J. C. Moses, <a href="/A261029/a261029.pdf">List of triples up to a(2000)</a>
%H A261029 Vladimir Shevelev, <a href="http://arxiv.org/abs/1508.05748">Representation of positive integers by the form x^3+y^3+z^3-3xyz</a>, arXiv:1508.05748 [math.NT], 2015.
%F A261029 For positive n, a(n)=0, if and only if n == 3 or 6 (mod 9); if p is prime, other than 3, then a(p) = a(2*p) = 1.
%F A261029 For n >= 1, a(8^(n-1)) = n.
%t A261029 r[n_] := Reduce[0 <= x <= y <= z && z >= x+1 && n == x^3 + y^3 + z^3 - 3 x y z, {x, y, z}, Integers];
%t A261029 a[n_] := Which[rn = r[n]; rn === False, 0, rn[[0]] === And, 1, rn[[0]] === Or, Length[rn], True, Print["error ", rn]];
%t A261029 Array[a, 100, 0] (* _Jean-François Alcover_, Nov 06 2018 *)
%Y A261029 Cf. A072670, A074232, A260803, A260804, A260965.
%K A261029 nonn
%O A261029 0,9
%A A261029 _Vladimir Shevelev_, Aug 22 2015
%E A261029 More terms from _Peter J. C. Moses_, Aug 22 2015