A350862 -id:A350862 - cp's OEIS frontend

A351885 Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).

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

Offset: 0

Daniel Hoyt, Feb 23 2022

The limiting difference between the integral and sum of x^(1/x). The limit converges slowly.

			0.5681800123590664525123147265218830744...

Daniel Hoyt, Graphical representation of the limit.
Daniel Hoyt, Computing the limiting difference between the sum and integral of x^(1/x).
Wikipedia, Stieltjes Constants

Cf. A350862, A329117, A175999, A001620.

# Gives 15 correct digits
from mpmath import stieltjes,fac,quad
def limgen(n):
    terms = []
    for y in range(3, n):
        for x in range(y, n):
            terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))
    return terms
f = lambda x: x**(1/x)
int01 = quad(f, [0,1])
limit = sum(limgen(60)) + 1.5 - stieltjes(0) - int01
print(limit)

Equals 3/2 - A001620 - A175999 + Sum_{k>=3} Sum_{n>=k} (((-1)^k)*Stieltjes(n)*(n-k+1)^(k-2))/((n-k+2)!*(k-2)!).

A363704 Decimal expansion of lim_{x -> infinity} ((Sum_{k>=1} (k^(1/k^(1 + 1/x)) - 1)) - x^2).

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Jun 16 2023

That this constant is less than one allows Sum_{k>=1} (k^(1/k^(1 + 1/x)) - 1) = floor(x^2), when x is the square root of any natural number greater than 1.

The limit converges slowly.

			0.98854960114226875064475410833997126442199868380...

Daniel Hoyt, Computing the limiting difference between the sum and integral of x^(1/x), 2022, pp. 1-4.
Wikipedia, Stieltjes Constants.

Cf. A098572, A350862, A329117, A351885, A001620, A363716.

digits = 120; d = 1; j = 2; s = StieltjesGamma[1]; While[Abs[d] > 10^(-digits - 5), d = (-1)^j / j! * Derivative[j][Zeta][j]; s += d; j++]; RealDigits[s, 10, 120][[1]] (* Vaclav Kotesovec, Jun 17 2023 *)

# Gives 14 correct digits
from mpmath import stieltjes,fac
def limgen(n):
    terms = []
    for y in range(3, n):
        for x in range(y, n):
            terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))
    n,o_sum = 2,0
    while True:
        n_term = 1/((n-1)**(n+1))
        n_sum = o_sum + n_term
        if o_sum == n_sum:
            break
        o_sum = n_sum
        n += 1
    return sum(terms) + 0.5 - stieltjes(0) + n_sum
print(str(limgen(60))[:-1])

Equals 1/2 - A001620 + Sum_{k>=2} (1/(k-1)^(k+1)) + Sum_{k>=3} Sum_{n>=k} (((-1)^k)*Stieltjes(n)*(n-k+1)^(k-2))/((n-k+2)!*(k-2)!).
From Vaclav Kotesovec, Jun 17 2023: (Start)
Equals lim_{n->oo} (Sum_{m=1..n} m^(1/m)) - n - log(n)^2/2.
Equals sg1 + Sum_{k>=2} (-1)^k / k! * k-th derivative of zeta(k), where sg1 is the first Stieltjes constant (see A082633). (End)

cp's OEIS Frontend

A351885 Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).

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