A364168 Numbers that can be written in more than one way in the form (j+2k)^2-(j+k)^2-j^2 with j,k>0.

15, 27, 32, 35, 36, 39, 51, 55, 60, 63, 64, 75, 84, 87, 91, 95, 96, 99, 100, 108, 111, 115, 119, 123, 128, 132, 135, 140, 143, 144, 147, 155, 156, 159, 160, 171, 175, 180, 183, 187, 192, 195, 196, 203, 204, 207, 215, 219, 220, 224, 228, 231, 235, 240, 243, 247, 252, 255

Offset

1,1

Links

Table of n, a(n) for n=1..58.

Project Euler, Problem 135: Same Differences.

Example

27 is a term since (6+2*3)^2 - (6+3)^2 - 6^2 = (20+2*7)^2 - (20+7)^2 - 20^2 = 27.

Prog

(Python)
def g(L):
  D = {}
  SQ = [x*x for x in range(0, 3*L+1)]
  for j in range(1, L):
    for k in range(1, L):
      q = SQ[j + k * 2] - SQ[j + k] - SQ[j]
      if (0 < q < L):
        if q not in D:
          D[q] = 1
        else:
          D[q] += 1
  return [x for x in sorted(D.keys()) if D[x] > 1]
print(g(256))
(Python)
from sympy import divisors
def isok(A_n):
  s = 0
  for d in divisors(A_n):
    t = A_n // d + d
    q, r = divmod(t, 4)
    if r == 0 and q < d: s += 1
  return s > 1

Crossrefs

Cf. A000290, A027750, A000005.

Sequence in context: A359205 A080945 A080946 * A253056 A091236 A354714

Adjacent sequences: A364165 A364166 A364167 * A364169 A364170 A364171

Keyword

nonn

Author

Darío Clavijo, Jul 12 2023

Status

approved