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A375618 a(n) is the least positive integer k such that there are n partitions k = x + y + z of positive integers such that x * y * z is a perfect cube or -1 if no such positive integer exists.

%I A375618 #21 Aug 22 2025 20:20:41
%S A375618 1,3,20,21,57,94,133,219,217,273,453,434,551,589,399,791,665,893,1321,
%T A375618 779,1330,1387,1519,1749,1786,2033,1767,2527,2793,1995,4066,3325,4389,
%U A375618 5548,4557,3895,4123,5187,5890,5529,5453,8075,6213,7980,7581,7790,11275,8113,11324,9310
%N A375618 a(n) is the least positive integer k such that there are n partitions k = x + y + z of positive integers such that x * y * z is a perfect cube or -1 if no such positive integer exists.
%H A375618 David A. Corneth, <a href="/A375618/b375618.txt">Table of n, a(n) for n = 0..303</a>
%H A375618 David A. Corneth, <a href="/A375618/a375618_1.gp.txt">PARI program</a>
%e A375618 a(1) = 3 as 3 = 1 + 1 + 1 and 1 * 1 * 1 = 1 is a perfect cube.
%p A375618 N:= 2*10^4:
%p A375618 V:= Array(1..N): count:= 0:
%p A375618 for x from 1 to N/3 do
%p A375618   for y from x to (N-x)/2 do
%p A375618      F:= ifactors(x*y)[2];
%p A375618      b:= mul(t[1],t = select(s -> s[2] mod 3 = 2, F));
%p A375618      c:= mul(t[1],t = select(s -> s[2] mod 3 = 1, F));
%p A375618      for k from ceil((y/(b*c^2))^(1/3)) do
%p A375618        s:= x+y+k^3 * b * c^2;
%p A375618        if s > N then break fi;
%p A375618        if s < x + 2*y then next fi;
%p A375618        V[s]:= V[s]+1
%p A375618 od od od:
%p A375618 m:= max(V):
%p A375618 A:= Array(0..m): A[0]:= 1: count:= 1:
%p A375618 for i from 1 to N while count < m+1 do
%p A375618   v:= V[i];
%p A375618   if A[v] = 0 then A[v]:= i; count:= count+1 fi
%p A375618 od:
%p A375618 AL:= convert(V,list);
%p A375618 if not member(0,AL,'r') then r:= m+2 fi;
%p A375618 AL[1..r-1]; # _Robert Israel_,  Oct 21 2024, corrected Aug 22 2025
%o A375618 (PARI) \\ See Corneth link
%Y A375618 Cf. A375580.
%K A375618 nonn
%O A375618 0,2
%A A375618 _David A. Corneth_, Aug 21 2024