A238489 Numbers k such that k+x+y is a square, where x and y are the two squares nearest to k.

0, 4, 11, 23, 56, 80, 103, 135, 204, 248, 339, 395, 444, 508, 576, 635, 711, 860, 948, 1119, 1219, 1304, 1412, 1619, 1739, 1968, 2100, 2211, 2351, 2495, 2616, 2768, 3055, 3219, 3528, 3704, 3851, 4035, 4223, 4380, 4576, 4943, 5151, 5540, 5760, 5943, 6171, 6596, 6836, 7283

Offset: 1

Author

Alex Ratushnyak, Feb 27 2014

Keywords

• nonn

Comments

If k is a square then y=k.

The sequence of terms that are perfect squares begins: 0, 4, 576, 108900, 21086464, 4090114116, 793453377600.

In other words, numbers x such that x + 2y(y+1) = z^2 has a solution with x in the interval [y^2+1, (y+1)^2], see Sage program. - Ralf Stephan, Mar 09 2014

The nonzero terms which are perfect squares are exactly the squares of A081065. - Ivan Neretin, Jun 25 2015

Examples

			The two squares nearest to 4 are 1 and 4. Because 4+1+4=9 is a square, 4 is in the sequence.
The two squares nearest to 11 are 9 and 16. Because 11+9+16=36 is a square, 11 is in the sequence.
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000290, A238456.

Programs

Mathematica

kxyQ[n_]:=Module[{c=Floor[Sqrt[n]]},IntegerQ[Sqrt[n+Total[Nearest[Range[c-2, c+2]^2,n,2]]]]]; Join[{0},Select[Range[3,7500],kxyQ]] (* Harvey P. Dale, Apr 24 2022 *)

Python

# use version >= 3.8
from math import isqrt
for k in range(7777):
    s = isqrt(k)
    if s*s==k:  s-=1
    kxy = k + 2*s*(s+1) + 1   # k + s^2 + (s+1)^2
    r = isqrt(kxy)
    if r*r==kxy:  print(str(k), end=',')

Sage

def gen_a():
    n = 1
    while True:
        for t in range(n*n + 1, n*n + 2*n + 2):
            if is_square(t + 2*(n*n + n) + 1):
                yield t
    n += 1              # Ralf Stephan, Mar 09 2014