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

3, 7, 8, 13, 17, 18, 21, 30, 31, 32, 41, 43, 46, 47, 50, 55, 57, 68, 72, 73, 75, 76, 83, 91, 93, 98, 99, 100, 105, 111, 112, 117, 119, 122, 123, 128, 129, 132, 133, 142, 144, 155, 157, 162, 172, 173, 174, 177, 182, 183, 185, 187, 189, 192, 193, 200, 203, 211

Offset: 1

Author

Matthijs Coster, Dec 17 2014

Keywords

• nonn

Comments

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.

Examples

			8 is in the sequence since arctan(1/8) = arctan(1/3) - arctan(1/5)
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

N:= 1000: # to get all terms <= N
A:= {}:
for x from 1 to N/2 do
   ds:= select(d -> (d <= x and d >= (x^2+1)/(N-x)), numtheory:-divisors(x^2+1));
   A:= A union map(d -> x + (x^2+1)/d, ds);
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list));
# Robert Israel, Dec 19 2014

SageMath

S = []
bound = 50
for b in range(1, bound-1):
    bb = b*b+1
    for d in divisors(bb):
        if (2*b < d) & (d-b < 2*bound):
            c = d-b
            a = (b*c-1)/(b+c)
            S.append((c, b, a))
S.sort()
print(S)