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

1,3

Comments

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

Links

Table of n, a(n) for n=1..30.

Wikipedia, Gamma Extrema

Wikipedia, Digamma Roots

Formula

A377376(a(n)) = n.

Example

 -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

Prog

(Python)
from mpmath import findroot, digamma, mp
def generate_sequence(n):
    i, j, seq, gamma_extrema = 0, mp.mpf(1), [], 1.4616321449683622
    while True:
        gamma_extrema = findroot(lambda x: digamma(x+1), x0=gamma_extrema-1)
        gamma_extrema = -mp.fabs(gamma_extrema)
        assert -i-1 < gamma_extrema < -i-0.4
        if mp.frac(gamma_extrema) < 1/j:
            seq.append(i)
            if j == n:
                break
            j += 1
        i += 1
    return seq
A374856 = generate_sequence(10)

Crossrefs

Cf. A030169, A175472, A175473, A377376.

Sequence in context: A038797 A136530 A259033 * A054459 A060188 A058751

Adjacent sequences: A374853 A374854 A374855 * A374857 A374858 A374859

Keyword

nonn,changed

Author

Jwalin Bhatt, Sep 16 2024

Status

approved