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.

1, 3, 20, 21, 57, 94, 133, 219, 217, 273, 453, 434, 551, 589, 399, 791, 665, 893, 1321, 779, 1330, 1387, 1519, 1749, 1786, 2033, 1767, 2527, 2793, 1995, 4066, 3325, 4389, 5548, 4557, 3895, 4123, 5187, 5890, 5529, 5453, 8075, 6213, 7980, 7581, 7790, 11275, 8113, 11324, 9310

Offset: 0

Author

David A. Corneth, Aug 21 2024

Keywords

• nonn

Examples

			a(1) = 3 as 3 = 1 + 1 + 1 and 1 * 1 * 1 = 1 is a perfect cube.
		

Links

• David A. Corneth, Table of n, a(n) for n = 0..303
• David A. Corneth, PARI program

Crossrefs

Cf. A375580.

Programs

Maple

N:= 2*10^4:
V:= Array(1..N): count:= 0:
for x from 1 to N/3 do
  for y from x to (N-x)/2 do
     F:= ifactors(x*y)[2];
     b:= mul(t[1],t = select(s -> s[2] mod 3 = 2, F));
     c:= mul(t[1],t = select(s -> s[2] mod 3 = 1, F));
     for k from ceil((y/(b*c^2))^(1/3)) do
       s:= x+y+k^3 * b * c^2;
       if s > N then break fi;
       if s < x + 2*y then next fi;
       V[s]:= V[s]+1
od od od:
m:= max(V):
A:= Array(0..m): A[0]:= 1: count:= 1:
for i from 1 to N while count < m+1 do
  v:= V[i];
  if A[v] = 0 then A[v]:= i; count:= count+1 fi
od:
AL:= convert(V,list);
if not member(0,AL,'r') then r:= m+2 fi;
AL[1..r-1]; # Robert Israel,  Oct 21 2024, corrected Aug 22 2025

PARI

\\ See Corneth link