A385170 a(n) is the integer part of the reciprocal of the distance of x_n from its nearest integer, where x_n is the n-th extrema of gamma(x).

1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Author

Jwalin Bhatt, Jun 20 2025

Keywords

• nonn

Comments

For n>=1, the nearest integer is -n and x_n drifts slowly towards it so that a(n) is a slowly increasing function of n.

a(n) = k occurs for the first time at n = A374856(k).

a(n) approximately equals floor(Pi/arctan(Pi/log(n))).

The corresponding gamma function value y_n = gamma(x_n) decreases in absolute value and A377506 captures its reciprocal (with a round).

Examples

			For n=10, x_10 = -9.702672... (A256687) and its nearest integer is -10 which is distance d = -9.702672... - (-10) = 0.297... away and a(n) = floor(1/d) = 3.
		

Links

• Jwalin Bhatt, Table of n, a(n) for n = 0..10000
• Wikipedia, Gamma Extrema
• Wikipedia, Digamma Roots

Crossrefs

Cf. A256687, A374856, A377506.

Programs

Mathematica

a[n_] := Module[
{bounds, gammaExtrema, fractionalPart, intReci},
bounds = {-n - 1 + 1/(n + 3), -n - 0.5};
gammaExtrema = x /. FindRoot[PolyGamma[0, x + 1] == 0, {x, Sequence @@ bounds}];
gammaExtrema = -Abs[gammaExtrema];
fractionalPart = Mod[gammaExtrema, 1];
Floor[1 / fractionalPart]
]

Python

from gmpy2 import mpq, get_context, 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])
        sign_d = sign(d)
        if not(is_nan(d)) and not(is_nan(u)) and (sign_d == sign(u)):
            return sign_d
        ctx.precision += 1
def generate_sequence(n):
    i, f, seq = -1, mpq('2'), [1]
    while len(seq) < n:
        if digamma_sign_near_int(i, f) != -1:
            f += 1
        seq.append(int(f)-1)
        i -= 1
    return seq
ctx = get_context()
ctx.precision = 2
A385170 = generate_sequence(101)