A214750 Least m > 0 such that n - m divides n^2 + m^2.

1, 1, 2, 3, 2, 5, 4, 3, 2, 9, 3, 11, 6, 5, 8, 15, 6, 17, 4, 3, 11, 21, 6, 15, 13, 9, 12, 27, 5, 29, 16, 11, 17, 10, 4, 35, 19, 13, 8, 39, 6, 41, 12, 15, 23, 45, 12, 35, 10, 17, 20, 51, 18, 5, 7, 19, 29, 57, 10, 59, 31, 9, 32, 15, 22, 65, 34, 23, 14, 69, 8, 71, 37, 25, 38

Offset: 2

Author

Clark Kimberling, Jul 29 2012

Keywords

• nonn
• easy

Comments

It appears that this is the sequence of k's for A110357. - Michel Marcus, Aug 16 2019

If n-m = s, then n = s+m and n-m | n^2+m^2 is equivalent to s | (s^2 + 2*s*m + 2*m^2). So n-m | n^2+m^2 is equivalent to n-m | 2*m^2. If n-k = s, then n = s+k and n-k | n*(n+k) is equivalent to s | (s^2 + 3*s*k + 2*k^2). So n-k | n*(n+k) is equivalent to n-k | 2*k^2. Therefore n-m | n^2+m^2 is equivalent to n-k | n*(n+k) and the k's from A110357 and the m's from this sequence are the same. - Bob Andriesse, Dec 26 2022

Let n-m = s; then m = n-s and n-m | n^2 + m^2 is equivalent to s | n^2 + (n-s)^2 or s | 2*n^2. If n is an odd prime, s must be 2. So if n is an odd prime, a(n) = m = n-2. Examples: a(7) = 5, a(11) = 9. - Bob Andriesse, Jul 13 2023

Examples

			Write x#y if x|y is false; then 7#65, 6#68, 5#73, 4|80, so a(8) = 4.
For n = 11, A110357(11) = 110 and a(11) = H(11, 110) - 11 = 20 - 11 = 9. - _Bob Andriesse_, Jan 03 2023
		

Links

• Clark Kimberling, Table of n, a(n) for n = 2..1000

Crossrefs

Cf. A110357, A214749.

Programs

Mathematica

Table[m = 1; While[! Divisible[n^2+m^2,n-m], m++]; m, {n, 2, 100}]

PARI

a(n) = my(m=1); while(denominator((n^2+m^2)/(n-m)) != 1, m++); m; \\ Michel Marcus, Aug 16 2019

Python

from sympy.abc import x, y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A214750(n): return min(int(x) for x,y in diop_quadratic(n*(n-y)+x*(y+x)) if x>0) # Chai Wah Wu, Oct 06 2023

Formula

a(n) = H(n, A110357(n)) - n where H is the harmonic mean. - Bob Andriesse, Jan 03 2023