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

1,2

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000290, A238456.

Sequence in context: A008252 A182707 A022495 * A002537 A295001 A230150

Adjacent sequences: A238486 A238487 A238488 * A238490 A238491 A238492

Keyword

nonn

Author

Alex Ratushnyak, Feb 27 2014

Status

approved