A232175 Least positive k such that n^3 + k^2 is a square, or 0 if there is no such k.

0, 1, 3, 6, 10, 3, 21, 8, 36, 15, 55, 6, 78, 35, 15, 48, 136, 27, 171, 10, 42, 99, 253, 10, 300, 143, 81, 42, 406, 15, 465, 64, 88, 255, 35, 63, 666, 323, 91, 3, 820, 21, 903, 55, 66, 483, 1081, 48, 1176, 125, 85, 39, 1378, 81, 165, 28, 76, 783, 1711, 15, 1830, 899, 63

Offset

1,3

Comments

Numbers n such that a(n) = n*(n-1)/2 appear to be A000430.

n = 1 is the only number for which a(n) = 0. - T. D. Noe, Nov 21 2013

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1000 from T. D. Noe).

StackExchange, The cube of integer can be written as the difference of two square.

Mathematica

Join[{0}, Table[k = 1; While[! IntegerQ[Sqrt[n^3 + k^2]], k++]; k, {n, 2, 100}]] (* T. D. Noe, Nov 21 2013 *)

Prog

(Python)
import math
for n in range(77):
   n3 = n*n*n
   y=1
   for k in range(1, 10000001):
     s = n3 + k*k
     r = int(math.sqrt(s))
     if r*r == s:
       print(k, end=', ')
       y=0
       break
   if y: print(end='-, ')
(Python)
from __future__ import division
from sympy import divisors
def A232175(n):
    n3 = n**3
    ds = divisors(n3)
    for i in range(len(ds)//2-1, -1, -1):
        x = ds[i]
        y = n3//x
        a, b = divmod(y-x, 2)
        if not b:
            return a
    return 0 # Chai Wah Wu, Sep 12 2017
(PARI) a(n) = {k = 1; while (!issquare(n^3+k^2), k++); k; } \\ Michel Marcus, Nov 20 2013

Crossrefs

Cf. A000290, A000430, A000578, A038202, A055527, A232176.

Sequence in context: A333611 A329153 A337771 * A065234 A333531 A082184

Adjacent sequences: A232172 A232173 A232174 * A232176 A232177 A232178

Keyword

nonn,look

Author

Alex Ratushnyak, Nov 19 2013

Status

approved