A238599 Numbers k such that k+x+y is a perfect cube, where x and y are the two cubes nearest to k.

0, 29, 171, 476, 1015, 1044, 1907, 3142, 4815, 7093, 9882, 13313, 17452, 22580, 28393, 35118, 42821, 43120, 51939, 61874, 72991, 85835, 99604, 114759, 131366, 150192, 170009, 191482, 214677, 240625, 267588, 296477, 327358, 361568, 396775, 434178, 473843, 475306, 517455

Offset

1,2

Links

Table of n, a(n) for n=1..39.

Example

The two cubes nearest to 0 are 0 and 1, and, because 0+0+1 is a perfect cube, 0 is in the sequence.
The two cubes nearest to 1 are 0 and 1, and, because 1+0+1=2 is not a perfect cube, 1 is not in the sequence.
The two cubes nearest to 29 are 27 and 8, and, because 29+27+8=64=4^3 is a perfect cube, 29 is in the sequence.

Mathematica

pcQ[n_]:=Module[{cr=Surd[n, 3]}, IntegerQ[Surd[Total[Nearest[Range[ Floor[ cr]-1, Ceiling[cr]+1]^3, n, 2]]+n, 3]]]; Select[Range[0, 520000], pcQ] (* Harvey P. Dale, Jul 25 2018 *)

Prog

(Python)
def icbrt(a):
    sr = 1 << (int.bit_length(int(a)) >> 1)
    while a < sr*sr*sr:  sr>>=1
    b = sr>>1
    while b:
        s = sr + b
        if a >= s*s*s:  sr = s
        b>>=1
    return sr
for k in range(1000000):
    s = icbrt(k)
    if k and s*s*s==k:  s-=1
    d1 = abs(k-s**3)
    d2 = abs(k-(s+1)**3)
    d3 = abs(k-(s-1)**3)
    kxy = k + s**3 + (s+1)**3
    if s and d3<d2:  kxy = k + s**3 + (s-1)**3
    r = icbrt(kxy)
    if r*r*r==kxy:  print(str(k), end=', ')
(Sage)
def gen_a():
    n = 1
    while True:
        for t in range(n*(n*n + 3), (n+1)*(n*n + 2*n + 4) + 1):
            c = t + (2*n + 1)*(n*n + n + 1)
            if round(floor(c^(1/3)))^3 == c:
                yield t
        n += 1               # Ralf Stephan, Mar 09 2014

Crossrefs

Cf. A000578, A238489.

Sequence in context: A264530 A141910 A111032 * A033219 A142407 A183714

Adjacent sequences: A238596 A238597 A238598 * A238600 A238601 A238602

Keyword

nonn

Author

Alex Ratushnyak, Mar 01 2014

Status

approved