A357945 Numbers k which are not square but D = (b+c)^2 - k is square, where b = floor(sqrt(k)) and c = k - b^2.

5, 13, 28, 65, 69, 76, 125, 128, 189, 205, 300, 305, 325, 352, 413, 425, 532, 533, 544, 565, 693, 725, 793, 828, 860, 1025, 1036, 1045, 1105, 1141, 1248, 1449, 1469, 1504, 1525, 1708, 1885, 1917, 1965, 2125, 2240, 2353, 2380, 2501, 2533, 2548, 2812, 2816, 2825, 2829, 2844, 2873, 2893

Offset: 1

Author

Darío Clavijo, Oct 21 2022

Keywords

• nonn

Comments

All composite terms are included in A177713.

Terms are the difference of two perfect squares k = (b+c)^2 - d^2, where d = sqrt(D), and so if composite are factorizable by Fermat's method k = ((b+c) + d) * ((b+c) - d).

Examples

			8525 is a term since it's not square and b = floor(sqrt(k)) = 92 and c = k - b^2 = 61 gives D = (b+c)^2 - k = 14884 which is square (122^2).
		

Crossrefs

Subsequence of A042965 and of A000037.

A211412 is a subsequence.

Cf. A053186, A177713.

Programs

PARI

isok(k) = if (!issquare(k), my(b=sqrtint(k), c=k-b^2); issquare((b+c)^2 - k)); \\ Michel Marcus, Oct 23 2022

Python

from gmpy2 import *
def is_A357945(n):
  if not is_square(n):
    b,c = isqrt_rem(n)
    return is_square(c*(2*b+c-1))
  else:
    return False

Formula

1.6*n < a(n) <= 4n^4 + 1. (Improvements welcome!) - Charles R Greathouse IV, Oct 23 2022