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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.

%I A246869 #19 Nov 13 2020 02:32:46
%S A246869 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,
%T A246869 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,
%U A246869 4,5,5,3,4,4,3,4,4,11,5,8,4,3,3,3,4,4
%N 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.
%C A246869 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.
%C A246869 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.
%C A246869 If there is no partition of n into four cubes, then a(n) = -1.
%C A246869 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
%H A246869 Alois P. Heinz, <a href="/A246869/b246869.txt">Table of n, a(n) for n = 0..20000</a>
%e A246869 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.
%p A246869 b:= proc(n, i, t) n=0 or t*i^3>=n and (b(n, i-1, t)
%p A246869       or b(n+i^3, i, t-1) or b(abs(n-i^3), i, t-1))
%p A246869     end:
%p A246869 a:= proc(n) local k; for k from 0
%p A246869       do if b(n, k, 4) then return k fi od
%p A246869     end:
%p A246869 seq(a(n), n=0..30);  # _Alois P. Heinz_, Sep 05 2014
%t A246869 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]);
%t A246869 a[n_] := Module[{k}, For[k = 0, True, k++, If[b[n, k, 4], Return[k]]]];
%t A246869 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 13 2020, after _Alois P. Heinz_ *)
%Y A246869 Cf. A243113.
%K A246869 nonn
%O A246869 0,6
%A A246869 _David S. Newman_, Sep 05 2014
%E A246869 More terms from _Alois P. Heinz_, Sep 05 2014