mini challenge: find the smallest cosine of an integer

[Albert Chan]

(10-21-2021 09:44 AM)EdS2 Wrote:  A004112 Numbers k where |cos(k)| (or |cosec(k)| or |cot(k)|) decreases monotonically to 0

Since it is impossible to input x to infinite precision, |cos(x)| does not decrease to zero.
With limited precision, |cos(x)| is not much smaller than machine-epsilon.

Quote:cos(24129) ≈ 0.002375894172
cos(24484) ≈ -0.002345749902
cos(40459) ≈ 0.000989256

FYI, all 3 x's comes from semi-convergents of pi/2:

344/219
355/226
51819/32989

24129 = 344 + 355*67
24484 = 344 + 355*68
40459 = 344 + 355*113

Quote:I wonder if the OEIS would like to add a series for the numerators of pi/2 convergents where the denominators are odd.

No need. Just use convergents of pi instead

pi/2 ≈ x / (2k+1)
pi ≈ (2x) / (2k+1)

With convergents fully reduced, even numerator guaranteed odd denominator.
So, just pick convergents/semi-convergents of pi, with even numerator = 2x

Note: convergents of pi/2 may have shifted to semi-convergents of pi.
Example, convergents of pi: 333/106, 355/113

(333+355) / (106+113) = 2*(344/219)       // (344/219) is a convergent of pi/2