A389852 Least positive integer d such that n + d^3 is the sum of 3 positive cubes.

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, 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, 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

Offset

1,1

Comments

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

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

Links

Table of n, a(n) for n=1..90.

Eric Weisstein's World of Mathematics, Cubic Number.

Eric Weisstein's World of Mathematics, Waring's Problem.

Example

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

Mathematica

f[n_]:=Module[{d, T, a, b, c, maxd=400}, For[d=1, d<=maxd, d++, T=n+d^3;
c=Floor[T^(1/3)];
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)];
If[a>0&&a<=b&&a^3+b^3+c^3==T, Return[d]]];
c--]]]; Table[f[n], {n, 1, 50}]

Crossrefs

Cf. A003072.

Sequence in context: A092082 A013559 A051186 * A373389 A012034 A138324

Adjacent sequences: A389849 A389850 A389851 * A389853 A389854 A389855

Keyword

nonn,easy

Author

Zhining Yang, Jan 23 2026

Status

approved