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A351885
Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).
3
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
0,1
COMMENTS
The limiting difference between the integral and sum of x^(1/x). The limit converges slowly.
FORMULA
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)!).
EXAMPLE
0.5681800123590664525123147265218830744...
PROG
(Python 3)
# 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)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Daniel Hoyt, Feb 23 2022
STATUS
approved