|
%I #35 Oct 06 2023 14:14:14
%S 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,
%T 29,16,11,17,10,4,35,19,13,8,39,6,41,12,15,23,45,12,35,10,17,20,51,18,
%U 5,7,19,29,57,10,59,31,9,32,15,22,65,34,23,14,69,8,71,37,25,38
%N Least m > 0 such that n - m divides n^2 + m^2.
%C It appears that this is the sequence of k's for A110357. - _Michel Marcus_, Aug 16 2019
%C 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
%C 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
%H Clark Kimberling, <a href="/A214750/b214750.txt">Table of n, a(n) for n = 2..1000</a>
%F a(n) = H(n, A110357(n)) - n where H is the harmonic mean. - _Bob Andriesse_, Jan 03 2023
%e Write x#y if x|y is false; then 7#65, 6#68, 5#73, 4|80, so a(8) = 4.
%e For n = 11, A110357(11) = 110 and a(11) = H(11, 110) - 11 = 20 - 11 = 9. - _Bob Andriesse_, Jan 03 2023
%t Table[m = 1; While[! Divisible[n^2+m^2,n-m], m++]; m, {n, 2, 100}]
%o (PARI) a(n) = my(m=1); while(denominator((n^2+m^2)/(n-m)) != 1, m++); m; \\ _Michel Marcus_, Aug 16 2019
%o (Python)
%o from sympy.abc import x, y
%o from sympy.solvers.diophantine.diophantine import diop_quadratic
%o 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
%Y Cf. A110357, A214749.
%K nonn,easy
%O 2,3
%A _Clark Kimberling_, Jul 29 2012
|
