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

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

1,1

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.

Links

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

Example

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

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

Prog

(SageMath)
S = []
    bound = 50
    for b in (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)

Crossrefs

Sequence in context: A010342 A108873 A002312 * A279517 A106474 A127441

Adjacent sequences: A252493 A252494 A252495 * A252497 A252498 A252499

Keyword

nonn

Author

Matthijs Coster, Dec 17 2014

Status

approved