A246869 - OEIS

A246869

Cube root of the smallest of the largest absolute values of parts of the partitions of n into four cubes, or -1 if no such partition exists.

0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 11, 2, 2, 2, 2, 2, 3, 3, 3, 16, 2, 2, 2, 3, 3, 3, 3, 3, 52, 2, 3, 3, 3, 3, 3, 3, 4, 4, 8, 3, 3, 3, 3, 3, 3, 4, 4, 49, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 3, 3, 3, 4, 4, 4, 4, 5, 5, 3, 4, 4, 3, 4, 4, 11, 5, 8, 4, 3, 3, 3, 4, 4

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OFFSET

0,6

COMMENTS

It is not known if every integer can be written as the sum of four cubes, but it is true at least up to 1000 by computer search.

For each partition of n into four cubes (positive, negative, or zero) choose the largest part in absolute value. a(n) is the cube root of the smallest such largest part over all such partitions.

If there is no partition of n into four cubes, then a(n) = -1.

There is an interesting correlation with A332201 (sum of three cubes problem) whose nonzero absolute values coincide with a(n+1) up to n=30. - M. F. Hasler, Feb 10 2020

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000

EXAMPLE

The partition of 13 into 1^3+7^3+10^3+(-11)^3 has a part 11^3 in absolute value. Any other partition of 13 into four cubes has a part larger than 11^3 in absolute value. Thus a(13) = 11.

MAPLE

b:= proc(n, i, t) n=0 or t*i^3>=n and (b(n, i-1, t)

or b(n+i^3, i, t-1) or b(abs(n-i^3), i, t-1))

end:

a:= proc(n) local k; for k from 0

do if b(n, k, 4) then return k fi od

end:

seq(a(n), n=0..30); # Alois P. Heinz, Sep 05 2014

MATHEMATICA

b[n_, i_, t_] := b[n, i, t] = n == 0 || t i^3 >= n && (b[n, i - 1, t] || b[n + i^3, i, t - 1] || b[Abs[n - i^3], i, t - 1]);

a[n_] := Module[{k}, For[k = 0, True, k++, If[b[n, k, 4], Return[k]]]];

a /@ Range[0, 100] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz *)