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A252496 Numbers k such that arctan(1/k) = arctan(1/x) - arctan(1/y) for some integers 0

%I A252496 #37 Jun 12 2025 16:49:32
%S A252496 3,7,8,13,17,18,21,30,31,32,41,43,46,47,50,55,57,68,72,73,75,76,83,91,
%T A252496 93,98,99,100,105,111,112,117,119,122,123,128,129,132,133,142,144,155,
%U A252496 157,162,172,173,174,177,182,183,185,187,189,192,193,200,203,211
%N A252496 Numbers k such that arctan(1/k) = arctan(1/x) - arctan(1/y) for some integers 0<x<y<k.
%C A252496 arctan(1/a(n)) = arctan(1/x) - arctan(1/y) for some integers x and y where 0 < x < y < a(n). We use the formula tan(a+b) = (tan a + tan b)/(1 - tan a.tan b) which implies that 1/a(n) = (1/x - 1/y)/(1+1/(xy)) or a(n) = (xy+1)/(y-x) = x + (x^2+1)/(y-x). So we look for divisors of x^2+1.
%H A252496 Robert Israel, <a href="/A252496/b252496.txt">Table of n, a(n) for n = 1..10000</a>
%e A252496 8 is in the sequence since arctan(1/8) = arctan(1/3) - arctan(1/5)
%p A252496 N:= 1000: # to get all terms <= N
%p A252496 A:= {}:
%p A252496 for x from 1 to N/2 do
%p A252496    ds:= select(d -> (d <= x and d >= (x^2+1)/(N-x)), numtheory:-divisors(x^2+1));
%p A252496    A:= A union map(d -> x + (x^2+1)/d, ds);
%p A252496 od:
%p A252496 A;
%p A252496 # if using Maple 11 or earlier, uncomment the next line
%p A252496 # sort(convert(A,list));
%p A252496 # _Robert Israel_, Dec 19 2014
%o A252496 (SageMath)
%o A252496 S = []
%o A252496 bound = 50
%o A252496 for b in range(1, bound-1):
%o A252496     bb = b*b+1
%o A252496     for d in divisors(bb):
%o A252496         if (2*b < d) & (d-b < 2*bound):
%o A252496             c = d-b
%o A252496             a = (b*c-1)/(b+c)
%o A252496             S.append((c, b, a))
%o A252496 S.sort()
%o A252496 print(S)
%K A252496 nonn
%O A252496 1,1
%A A252496 _Matthijs Coster_, Dec 17 2014