A371124 a(n) is the least nonnegative integer y such that y^2 = x^2 - k*n for k and x where n > k >= 1 and n > x >= floor(sqrt(n)).

0, 0, 1, 0, 2, 2, 3, 1, 0, 4, 5, 2, 6, 6, 1, 0, 8, 0, 9, 4, 2, 10, 11, 1, 0, 12, 3, 6, 14, 2, 15, 2, 4, 16, 1, 0, 18, 18, 5, 3, 20, 4, 21, 10, 2, 22, 23, 1, 0, 0, 7, 12, 26, 6, 3, 5, 8, 28, 29, 2, 30, 30, 1, 0, 4, 8, 33, 16, 10, 2, 35, 3, 36, 36, 5, 18, 2, 10

Offset

1,5

Comments

a(A000290(n)) = 0.

a(A077591(n)) = 0.

a(A005563(n)) = 1.

For each n: k = A138191(n) and x = A306284(n).

Links

Table of n, a(n) for n=1..78.

Formula

a(n) = floor(sqrt(A306284(n)^2 - n*A138191(n))).

a(A000040(n)) = A102781(n).

Example

 n  | k | x | y^2 = x^2 - k*n  | y
------------------------------------
 1  | 1 | 1 | 0^2 = 1^2 - 1*1  | 0
 2  | 2 | 2 | 0^2 = 2^2 - 2*1  | 0
 11 | 1 | 6 | 5^2 = 6^2 - 1*11 | 5

Prog

(Python)
from sympy.core.power import isqrt
from sympy.ntheory.primetest import is_square
def a(n):
  x = isqrt(n)
  while True:
    for y2 in range(x**2-n, -1, -n):
      if is_square(y2): return isqrt(y2)
    x+=1
print([a(n) for n in range(1, 79)])
(Python)
from itertools import count
def A371124(n):
    y, a = 0, {}
    for x in count(0):
        if y in a: return a[y]
        a[y] = x
        y = (y+(x<<1)+1)%n # Chai Wah Wu, Apr 25 2024

Crossrefs

Cf. A000040, A000290, A005563, A077591, A102781, A138191, A306284.

Sequence in context: A290563 A071474 A071480 * A267483 A071473 A084189

Adjacent sequences: A371121 A371122 A371123 * A371125 A371126 A371127

Keyword

nonn

Author

Darío Clavijo, Mar 11 2024

Status

approved