A374856 a(n) is the least integer m such that the distance of x_m from its nearest integer is less than 1/n, where x_m is the m-th extrema of Gamma(x).

0, 1, 6, 23, 76, 231, 681, 1968, 5605, 15817, 44324, 123573, 343157, 950000, 2623530, 7230746, 19896140, 54671729, 150058028, 411465352, 1127315946, 3086355718, 8444524052, 23092305853, 63117665557, 172444844373, 470961842866, 1285804853026, 3509404275438, 9575773901601

Offset: 1

Author

Jwalin Bhatt, Sep 16 2024

Keywords

• nonn

Comments

a(n) approximately equals exp(Pi/tan(Pi/n)).

Examples

			 -1 + 1.46163214496836 < 1/1
  1 - 0.50408300826446 < 1/2
  6 - 5.66716244155689 < 1/3
 23 - 22.7502429843061 < 1/4
 76 - 75.8003723367285 < 1/5
231 - 230.833395691244 < 1/6
		

Links

• Wikipedia, Gamma Extrema
• Wikipedia, Digamma Roots

Crossrefs

Cf. A030169, A175472, A175473, A377506, A385170.

Programs

Python

from gmpy2 import mpq, get_context, exp, digamma, sign, is_nan, RoundUp, RoundDown
def apply_on_interval(func, interval):
    ctx.round = RoundUp
    rounded_up = func(interval[0])
    ctx.round = RoundDown
    rounded_down = func(interval[1])
    return rounded_down, rounded_up
def digamma_sign_near_int(i, f):
    while True:
        d, u = apply_on_interval(lambda x: digamma(i + 1/x), [f, f])
        if not(is_nan(d)) and not(is_nan(u)) and (sign(d) == sign(u)):  return sign(d)
        ctx.precision += 1
def find_next_zero_crossing(f, i, growth_factor):  # Bisect.
    lo, hi = int(i * const_e), int(i * growth_factor)
    while lo - 1 != hi:
        if digamma_sign_near_int(mid := (hi + lo) // 2, f) == -1:  lo = mid
        else:  hi = mid
    return hi
def generate_sequence(n):
    seq, frac_denoms = [0, 1, 6], (mpq(str(i)) for i in range(4, n + 1))
    for f in frac_denoms:  seq.append(-find_next_zero_crossing(f, -seq[-1], seq[-1] / seq[-2]))
    return seq
const_e, ctx = exp(1), get_context()
ctx.precision = 2
A374856 = generate_sequence(30)

Formula

A385170(a(n)) = n.