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

Author

Alex Ratushnyak, Nov 19 2013

Keywords

• nonn
• look

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.

Crossrefs

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

Programs

Mathematica

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

PARI

a(n) = {k = 1; while (!issquare(n^3+k^2), k++); k;} \\ Michel Marcus, Nov 20 2013

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