A233422 Numbers n such that m - n^3 is a square, where m is the least square above n^3.

0, 2, 3, 6, 12, 20, 24, 30, 40, 42, 56, 60, 68, 75, 78, 84, 87, 120, 126, 160, 180, 248, 264, 270, 273, 308, 312, 318, 330, 336, 351, 360, 396, 564, 570, 588, 615, 620, 630, 635, 720, 738, 780, 840, 912, 1008, 1016, 1032, 1284, 1308, 1320, 1334, 1344, 1404, 1540, 1617

Offset

1,2

Comments

Numbers n such that A070929(n) is a nonzero square.

The sequence of cubes a(n)^3 begins: 0, 8, 27, 216, 1728, 8000, 13824, 27000, 64000, 74088, 175616, 216000, 314432, ...

The sequence of m's begins: 1, 9, 36, 225, 1764, 8100, 13924, 27225, 64009, 74529, 176400, 216225, 314721, ...

The sequence of square roots of these m's begins: 1, 3, 6, 15, 42, 90, 118, 165, 253, 273, 420, 465, 561, 650, 689, 770, 812, ...

The sequence of squares m-n^3 begins: 1, 1, 9, 9, 36, 100, 100, 225, 9, 441, 784, 225, 289, 625, 169, 196, 841, ...

The sequence of their square roots begins: 1, 1, 3, 3, 6, 10, 10, 15, 3, 21, 28, 15, 17, 25, 13, 14, 29, 35, 43, 24, ... (note the first 12 terms are triangular numbers, A000217).

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Mathematica

fQ[n_]:=Module[{c=n^3, m}, m=(Floor[Sqrt[c]]+1)^2; IntegerQ[Sqrt[m-c]]]; Select[Range[0, 1650], fQ] (* Harvey P. Dale, Jan 03 2024 *)

Prog

(Python)
def isqrt(a):
    sr = 1L << (long.bit_length(long(a)) >> 1)
    while a < sr*sr:  sr>>=1
    b = sr>>1
    while b:
      s = sr+b
      if a >= s*s:  sr = s
      b>>=1
    return sr
def isSquare(a):
  sr = isqrt(a)
  return (a==sr*sr)
for n in range(77777):
  n3 = n*n*n
  a = isqrt(n3)+1
  if isSquare(a*a-n3):  print str(n)+', ',
(PARI) is(n)=issquare((sqrtint(n=n^3)+1)^2-n) \\ Charles R Greathouse IV, Dec 09 2013

Crossrefs

Cf. A000290, A000578, A070929, A077115, A154101, A233400, A233401.

Sequence in context: A309311 A361693 A103070 * A361382 A079708 A096571

Adjacent sequences: A233419 A233420 A233421 * A233423 A233424 A233425

Keyword

nonn

Author

Alex Ratushnyak, Dec 09 2013

Status

approved