A359078 a(n) is the first positive number that can be represented in exactly n ways as the sum of cubes of three distinct integers in arithmetic progression.

9, 99, 792, 3829608, 255816, 24814152, 198513216, 1588105728, 669982104, 5359856832, 42878854656, 7133969443392, 57071755547136

Offset: 1

Author

Robert Israel, Dec 15 2022

Keywords

• nonn
• more

Comments

a(n) is the first positive number that can be represented in exactly n ways as 3*x*(x^2+2*y^2) = (x-y)^3 + x^3 + (x+y)^3 for positive numbers x and y.

Note that the first term x-y of the arithmetic progression need not be positive.

From Ondrej Kutal, Dec 21 2022: (Start)

a(14) <= 41605309793862144.

a(15) <= 35049191875384896000.

a(16) <= 25550860877155589184000. (End)

Examples

			a(3) = 792 because 792 = (-6)^3 + 2^3 + 10^3 = (-1)^3 + 4^3 + 9^3 = 4^3 + 6^3 + 8^3 is the first number that can be represented in exactly 3 ways.
		

Crossrefs

Cf. A359030.

Programs

Maple

N:= 5*10^10:
L:= NULL:
for a from 1 while 3*a^3 <= N do
  for b from 1 do
    x:= 3*a*(a^2 + 2*b^2);
    if x > N then break fi;
    L:= L,x
od od:
L:= sort([L]):
V:= Vector(11):
m:= L[1]: count:= 1:
for i from 2 to nops(L) do
if L[i] = m then count:= count+1
else
  if V[count] = 0 then V[count]:= m fi;
  count:= 1; m:= L[i];
fi
od:
convert(V,list);

Extensions

a(12)-a(13) from Ondrej Kutal, Dec 21 2022