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).

%I A351885 #33 May 22 2025 18:37:38
%S A351885 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,
%T A351885 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,
%U A351885 3,4,2,7,6,8,0,7,3,5,1,0,0,4,2,2,1,7,5
%N A351885 Decimal expansion of lim_{n -> infinity} (Sum_{x=1..n} x^(1/x) - Integral_{k=0..n} x^(1/x) dx).
%C A351885 The limiting difference between the integral and sum of x^(1/x). The limit converges slowly.
%H A351885 Daniel Hoyt, <a href="/A351885/a351885.png">Graphical representation of the limit.</a>
%H A351885 Daniel Hoyt, <a href="/A351885/a351885.pdf">Computing the limiting difference between the sum and integral of x^(1/x).</a>
%H A351885 Wikipedia, <a href="https://en.wikipedia.org/wiki/Stieltjes_constants">Stieltjes Constants</a>
%F A351885 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)!).
%e A351885 0.5681800123590664525123147265218830744...
%o A351885 (Python)
%o A351885 # Gives 15 correct digits
%o A351885 from mpmath import stieltjes,fac,quad
%o A351885 def limgen(n):
%o A351885     terms = []
%o A351885     for y in range(3, n):
%o A351885         for x in range(y, n):
%o A351885             terms.append((((-1)**y)*stieltjes(x)*(x-(y-1))**(y-2))/(fac(x-(y-2))*fac(y-2)))
%o A351885     return terms
%o A351885 f = lambda x: x**(1/x)
%o A351885 int01 = quad(f, [0,1])
%o A351885 limit = sum(limgen(60)) + 1.5 - stieltjes(0) - int01
%o A351885 print(limit)
%Y A351885 Cf. A350862, A329117, A175999, A001620.
%K A351885 nonn,cons
%O A351885 0,1
%A A351885 _Daniel Hoyt_, Feb 23 2022