A389852 - OEIS

%I #85 Jan 28 2026 19:09:27

%S 7,1,101,5,304,7,9,14,1,11,5,14,11,22,25,1,4,26,13,8,2,53,1,61,7,9,15,

%T 1,10,5,257,7,9,37,1,11,6,15,3,149,28,1,14,13,8,3,2,8,26,49,10,25,3,1,

%U 8,14,47,2,88,163,1,7,4,28,1,111,5,187,22,4,10,1,2,8,20,11,4,21,1,1,8,9,18,2,86,31,33,10,4,7

%N Least positive integer d such that n + d^3 is the sum of 3 positive cubes.

%C Conjecture: a(n) exists for all positive integers n.

%C For all n <= 10^6, a(493196) = 1858 is the largest.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CubicNumber.html">Cubic Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/WaringsProblem.html">Waring's Problem</a>.

%e a(3) = 101 because 3 + 101^3 = 18^3 + 70^3 + 88^3, and no integers less than 101 satisfied this way.

%t f[n_]:=Module[{d,T,a,b,c,maxd=400},For[d=1,d<=maxd,d++,T=n+d^3;

%t c=Floor[T^(1/3)];

%t While[c>=Ceiling[(T/3)^(1/3)],For[b=Min[c,Floor[(T-c^3)^(1/3)]],b>=Ceiling[((T-c^3)/2)^(1/3)],b--,a=Round[(T-c^3-b^3)^(1/3)];

%t If[a>0&&a<=b&&a^3+b^3+c^3==T,Return[d]]];

%t c--]]]; Table[f[n],{n,1,50}]

%Y Cf. A003072.

%K nonn,easy

%O 1,1

%A _Zhining Yang_, Jan 23 2026