A175473 -id:A175473 - cp's OEIS frontend

A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^3, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			-7.84898826280450630489883732716055067110164127911638...

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, A344966, A344967, A344968.
Cf. A001620, A002117, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[EulerGamma*Pi^2/2 + 4*Zeta[3]  + EulerGamma^3, 10, 100][[1]]

Equals -gamma*Pi^2/2 - 4*zeta(3) - gamma^3, where gamma is Euler's constant (A001620).

A344966 Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^4, where x_k is the k-th zero of the digamma function, i.e., root of psi(x) = 0: x_0 = 1.461632... (A030169) is the only positive root, x_1 = -0.504083... (A175472), etc.

			15.90184703322349156972084557358425176519256672643402...

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, A344967, A344968.
Cf. A000796, A001620, A002117.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^4/9 + 2*EulerGamma^2*Pi^2/3 + 4*EulerGamma*Zeta[3] + EulerGamma^4, 10, 100][[1]]

Equals Pi^4/9 + 2*gamma^2*Pi^2/3 + 4*gamma*zeta(3) + gamma^4, where gamma is Euler's constant (A001620).

A344967 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - 1), where x_k is the k-th zero of the digamma function.

Original entry on oeis.org

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

Offset: 0

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.

			0.71349472210996816769933594441333563665531893958512...

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, A344968.
Cf. A000796, A001620, A072691.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/(12*EulerGamma) + EulerGamma/2 - 1, 10, 100][[1]]

Equals Pi^2/(12*gamma) + gamma/2 - 1, where gamma is Euler's constant (A001620).

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.

Original entry on oeis.org

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).

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).

Original entry on oeis.org

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

Jwalin Bhatt, Sep 16 2024

nonn

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

			 -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

Wikipedia, Gamma Extrema
Wikipedia, Digamma Roots

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

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)

A385170(a(n)) = n.

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A344965 Decimal expansion of the sum of the reciprocals of the cubes of the zeros of the digamma function (negated).

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A344966 Decimal expansion of the sum of the reciprocals of the fourth powers of the zeros of the digamma function.

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A344967 Decimal expansion of Sum_{k>=0} 1/(x_k^2 - 1), where x_k is the k-th zero of the digamma function.

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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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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).

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