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A004825 Numbers that are the sum of at most 3 positive cubes.

%I A004825 #43 Oct 27 2023 08:19:44
%S A004825 0,1,2,3,8,9,10,16,17,24,27,28,29,35,36,43,54,55,62,64,65,66,72,73,80,
%T A004825 81,91,92,99,118,125,126,127,128,129,133,134,136,141,152,153,155,160,
%U A004825 179,189,190,192,197,216,217,218,224,225,232,243,244,250,251,253
%N A004825 Numbers that are the sum of at most 3 positive cubes.
%C A004825 Or: numbers which are the sum of 3 (not necessarily distinct) nonnegative cubes. - _R. J. Mathar_, Sep 09 2015
%C A004825 Deshouillers, Hennecart, & Landreau conjecture that this sequence has density 0.0999425... = lim_K Sum_{k=1..K} exp(c*rho(k,K)/K^2)/K where c = -gamma(4/3)^3/6 = -0.1186788..., K takes increasing values in A003418 (or, equivalently, A051451), and rho(k0,K) is the number of triples 1 <= k1,k2,k3 <= K such that k0 = k1^3 + k2^3 + k3^3 mod K. - _Charles R Greathouse IV_, Sep 16 2016
%H A004825 Alois P. Heinz, <a href="/A004825/b004825.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A004825 Jean-Marc Deshouillers, François Hennecart, and Bernard Landreau, <a href="http://link.springer.com/chapter/10.1007/11792086_11">On the density of sums of three cubes</a>, ANTS-VII (2006), pp. 141-155.
%H A004825 <a href="/index/Su#ssq">Index entries for sequences related to sums of cubes</a>
%p A004825 isA004825 := proc(n)
%p A004825     local x,y,zc ;
%p A004825     for x from 0 do
%p A004825         if 3*x^3 > n then
%p A004825             return false;
%p A004825         end if;
%p A004825         for y from x do
%p A004825             if x^3+2*y^3 > n then
%p A004825                 break;
%p A004825             else
%p A004825                 zc := n-x^3-y^3 ;
%p A004825                 if zc >= y^3 and isA000578(zc) then
%p A004825                     return true;
%p A004825                 end if;
%p A004825             end if;
%p A004825         end do:
%p A004825     end do:
%p A004825 end proc:
%p A004825 A004825 := proc(n)
%p A004825     option remember;
%p A004825     local a;
%p A004825     if n = 1 then
%p A004825         0;
%p A004825     else
%p A004825         for a from procname(n-1)+1 do
%p A004825             if isA004825(a) then
%p A004825                 return a;
%p A004825             end if;
%p A004825         end do:
%p A004825     end if;
%p A004825 end proc:
%p A004825 seq(A004825(n),n=1..100) ; # _R. J. Mathar_, Sep 09 2015
%p A004825 # second Maple program:
%p A004825 b:= proc(n, i, t) option remember; n=0 or i>0 and t>0
%p A004825       and (b(n, i-1, t) or i^3<=n and b(n-i^3, i, t-1))
%p A004825     end:
%p A004825 a:= proc(n) option remember; local k;
%p A004825       for k from 1+ `if`(n=1, -1, a(n-1))
%p A004825       while not b(k, iroot(k, 3), 3) do od; k
%p A004825     end:
%p A004825 seq(a(n), n=1..100);  # _Alois P. Heinz_, Sep 16 2016
%t A004825 q=7; imax=q^3; Select[Union[Flatten[Table[x^3+y^3+z^3, {x,0,q}, {y,x,q}, {z,y,q}]]], #<=imax&] (* _Vladimir Joseph Stephan Orlovsky_, Apr 20 2011 *)
%o A004825 (PARI) list(lim)=my(v=List(),k,t); for(x=0,sqrtnint(lim\=1,3), for(y=0, min(sqrtnint(lim-x^3,3),x), k=x^3+y^3; for(z=0,min(sqrtnint(lim-k,3), y), listput(v, k+z^3)))); Set(v) \\ _Charles R Greathouse IV_, Sep 14 2015
%Y A004825 A003072 is a subsequence.
%Y A004825 Cf. A004999.
%Y A004825 Column k=3 of A336820.
%K A004825 nonn
%O A004825 1,3
%A A004825 _N. J. A. Sloane_