A030169 -id:A030169 - cp's OEIS frontend

A256686 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-9,-8].

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

Offset: 1

Jean-François Alcover, Apr 08 2015

			Gamma(-8.695764163816401266488776160804645820272438084966728783...)
= -0.00002092529044652666875369728468060738117860083247673665...

Eric Weisstein's MathWorld, Gamma Function
Wikipedia, Particular values of the Gamma Function

Cf. A030169, A030171, A175472, A175473, A256681-A256685, A256687.

digits = 104; x0 = x /. FindRoot[PolyGamma[0, x] == 0, {x, -17/2}, WorkingPrecision -> digits + 5]; RealDigits[x0, 10, digits] // First

Solution to PolyGamma(x) = 0 in the interval [-9,-8].

A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 03 2021

The sum is Sum_{k>=0} 1/x_k^2, 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.

			5.26798012435239798373562163629331979431626684387002...

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. A344965, A344966, A344967, A344968.
Cf. A001620, A102753.
Cf. A030169, A175472, A175473, A175474, A256681, A256682, A256683, A256684, A256685, A256686, A256687.

RealDigits[Pi^2/2 + EulerGamma^2, 10, 100][[1]]

Equals Pi^2/2 + gamma^2 = A102753 + A155969, where gamma is Euler's constant (A001620).

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

Original entry on oeis.org

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.

A030170 Let c be the point at which Gamma(x), x>0, is minimized; sequence gives continued fraction for c.

Original entry on oeis.org

1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, 9, 2, 5, 734, 6, 4, 1, 1, 2, 16, 1, 1, 1, 2, 1, 5, 3, 1, 2, 2478, 1, 1, 1, 3, 8, 1, 1, 7, 1, 1, 1, 64, 29, 3, 10, 2, 5, 1, 1, 2, 61, 1, 39, 1, 5, 1, 1, 2, 1, 8, 1, 16, 12, 3, 1, 32, 1

Offset: 0

Eric W. Weisstein

G. C. Greubel, Table of n, a(n) for n = 0..5000
R. Schroeppel, HAKMEM 140 - Continued Fractions: ITEM 98 (Minimum of factorial function). M.I.T., Artificial Intelligence Memo No. 239, February 29, 1972.
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A030169.

ContinuedFraction[ x /. FindRoot[ Gamma'[x] == 0, {x, 1}, WorkingPrecision -> 100], 69] (* Jean-François Alcover, Oct 29 2012 *)

contfrac(solve(x=1, 2, psi(x))) \\ Michel Marcus, Oct 18 2017

A096188 Engel expansion of real number x such that y = Gamma(x) is a minimum.

Original entry on oeis.org

1, 3, 3, 7, 13, 14, 14, 27, 27, 46, 99, 549, 913, 2637, 3830, 3929, 15500, 55253, 85854, 246166, 1052057, 2490138, 2521393, 16086534, 29730193, 38774343, 84328391, 317160458, 371478595, 600277187, 811735945, 849656112, 139143919171

Offset: 1

Gerald McGarvey, Jul 25 2004

nonn

Gamma(x) has a minimum at x = 1.46163214496836234126265954232572132846819620400644... (A030169).

Xavier Gourdon and Pascal Sebah, Some Constants from Number theory from their "Numbers, constants and computation" web site.

Cf. A030169.

EngelExp[ A_, n_ ] := Join[ Array[ 1 &, Floor[ A ]], First@Transpose @ NestList[ {Ceiling[ 1/Expand[ #[[ 1 ]] #[[ 2 ]] - 1 ]], Expand[ #[[ 1 ]] #[[ 2 ]] - 1]} &, {Ceiling[ 1/(A - Floor[A]) ], A - Floor[A]}, n - 1 ]]; EngelExp[ FindMinimum[ Gamma[x], {x, 1, 4}, WorkingPrecision -> 2^9][[2, 1, 2]], 32] (* Robert G. Wilson v, Jul 28 2004 *)

Edited and extended by Robert G. Wilson v, Jul 28 2004

A365797 Decimal expansion of smallest positive number x such that Gamma(x) = 2.

Original entry on oeis.org

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

Offset: 0

David Ulgenes, Sep 19 2023

Second branch (i.e., the first after the principal branch) of the inverse gamma function Gamma(y) = x at x=2. See for instance Uchiyama.
Since 1 - x = 0.55712260351... (approximately equal to A249649), we can obtain the interesting approximation Gamma(zeta(2) - zeta(3)) ≈ 2.000001... - David Ulgenes, Feb 19 2024
x is the least positive real number where 1+Gamma(1+Gamma(1+Gamma...(x)...)) converges; it converges to 3. - Colin Linzer, Nov 25 2024

			0.4428773964847274374520325165206056717103645380663664...

K. Amenyo Folitse, David J. Jeffrey, and Robert M. Corless, Properties and Computation of the Functional Inverse of Gamma, 2017 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). p. 65.
Mitsuru Uchiyama, The principal inverse of the gamma function, Proc. Amer. Math. Soc. 140 (2012), 1343-1348.

Cf. A030169, A030171.

Digits:= 140:
with(RootFinding):
NextZero(x -> (x - 1)! - 2, 0);

FindRoot[-2 + (-1 + x)! == 0, {x, 0, 1}, WorkingPrecision -> 15]

solve(x=0.1, 1, gamma(x)-2) \\ Michel Marcus, Sep 19 2023

Equals ((((1/2)!/2)!/2)!/2)!/2...
Proof: Since y = y! / x we substitute the expression into itself to obtain an iterative scheme for the inverse gamma function.
Equals (1/(2*Pi))*Integral_{x=-oo..oo} log((2-Gamma(i*x))/(2-Gamma(1+i*x))) dx. Proof: Follows from writing the inverse gamma function using the Lagrange inversion theorem together with Cauchy's formula for differentiation. - David Ulgenes, Feb 11 2024

cp's OEIS Frontend

A256686 Decimal expansion of the [negated] abscissa of the Gamma function local maximum in the interval [-9,-8].

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A344964 Decimal expansion of the sum of the reciprocals of the squares of the zeros of the digamma function.

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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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A030170 Let c be the point at which Gamma(x), x>0, is minimized; sequence gives continued fraction for c.

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