A256687 -id:A256687 - cp's OEIS frontend

A344968 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - x_k), where x_k is the k-th zero of the digamma function.

3, 4, 2, 6, 9, 8, 9, 4, 4, 4, 2, 1, 9, 9, 3, 6, 3, 3, 5, 3, 9, 8, 6, 7, 1, 8, 8, 8, 8, 2, 6, 6, 7, 1, 2, 7, 3, 3, 1, 0, 6, 3, 7, 8, 7, 9, 1, 7, 0, 2, 5, 9, 0, 1, 1, 8, 9, 1, 7, 7, 4, 0, 3, 3, 1, 6, 2, 0, 9, 4, 4, 8, 1, 5, 8, 4, 3, 3, 7, 3, 7, 2, 1, 2, 3, 7, 4

Offset: 1

Amiram Eldar, Jun 03 2021

The zeros of the digamma function, i.e., the roots of psi(x) = 0 are x_0 = 1.461632... (A030169), the only positive root, x_1 = -0.504083... (A175472), etc.

			3.42698944421993633539867188882667127331063787917025...

István Mező and Michael E. Hoffman, Zeros of the digamma function and its Barnes G-function analogue, Integral Transforms and Special Functions, Vol. 28, No. 11 (2017), pp. 846-858.
Wikipedia, Digamma function.

Cf. A344964, A344965, A344966, A344967.
Cf. A001620, A013661.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/(6*EulerGamma) + EulerGamma, 10, 100][[1]]

Equals Pi^2/(6*gamma) + gamma, where gamma is Euler's constant (A001620).

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).

Original entry on oeis.org

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

Jwalin Bhatt, Jun 20 2025

nonn

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).

			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.

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

Cf. A256687, A374856, A377506.

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]
]

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)

cp's OEIS Frontend

A344968 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - x_k), where x_k is the k-th zero of the digamma function.

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