A233401 Numbers k such that k^3 - b2 is a triangular number (A000217), where b2 is the largest square less than k^3.

1, 4, 8, 21, 37, 40, 56, 112, 113, 204, 280, 445, 481, 560, 688, 709, 1933, 1945, 3601, 3805, 3861, 4156, 4333, 4365, 7096, 8408, 8516, 11064, 12688, 13609, 13945, 16501, 17080, 18901, 21464, 23125, 27244, 27364, 28141, 45228, 45549, 58321, 60061, 66245, 70585, 78688

Offset: 1

Author

Alex Ratushnyak, Dec 09 2013

Keywords

• nonn

Comments

The cubes k^3 begin: 1, 64, 512, 9261, 50653, 64000, 175616, 1404928, ...

The squares b2 begin: 0, 49, 484, 9216, 50625, 63504, 175561, 1404225, ...

Their square roots are 0, 7, 22, 96, 225, 252, 419, 1185, 1201, 2913, 4685, 9387, ...

Crossrefs

Cf. A000217, A000290, A000578, A048760, A233400.

Programs

PARI

f(k) = if (issquare(k), sqrtint(k-1)^2, sqrtint(k)^2); \\ adapted from A048760
isok(k) = my(b2 = sqrtint(k^3-1)^2); (k^3-b2) && ispolygonal(k^3-b2, 3); \\ Michel Marcus, Jan 26 2019

Python

from math import isqrt
def isTriangular(a):
  a+=a
  sr = isqrt(a)
  return (a==sr*(sr+1))
for n in range(1,79999):
  n3 = n*n*n
  b = isqrt(n3)
  if b*b==n3: b-=1
  if isTriangular(n3-b*b):  print(n, end=', ')