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A333072 Least k such that Sum_{i=1..n} k^i / i is a positive integer.

%I A333072 #15 Apr 04 2020 16:29:43
%S A333072 1,2,6,6,30,10,70,70,210,168,1848,1848,18018,8580,2574,2574,102102,
%T A333072 102102,831402,2771340,3233230,587860,43266496,117630786,162249360,
%U A333072 145088370,145088370,2897310,672175920,672175920,18232771830,18232771830,44279588730,8886561060
%N A333072 Least k such that Sum_{i=1..n} k^i / i is a positive integer.
%C A333072 Note that the denominator of (Sum_{i=1..n} k^i/i) - k^p/p can never be divisible by p, where n/2 < p <= n. Therefore, for the expression to be an integer, such p must divide k. Thus, a(n) = k is divisible by A055773(n).
%H A333072 Chai Wah Wu, <a href="/A333072/b333072.txt">Table of n, a(n) for n = 1..61</a>
%F A333072 a(n) <= A034386(n).
%o A333072 (PARI) a(n) = {my(m = prod(i=primepi(n/2)+1, primepi(n), prime(i)), k = m); while (denominator(sum(i=2, n, k^i/i)) != 1, k += m); k; }
%o A333072 (Python)
%o A333072 from sympy import primorial, lcm
%o A333072 def A333072(n):
%o A333072     f = 1
%o A333072     for i in range(1,n+1):
%o A333072         f = lcm(f,i)
%o A333072     f, glist = int(f), []
%o A333072     for i in range(1,n+1):
%o A333072         glist.append(f//i)
%o A333072     m = 1 if n < 2 else primorial(n,nth=False)//primorial(n//2,nth=False)
%o A333072     k = m
%o A333072     while True:
%o A333072         p,ki = 0, k
%o A333072         for i in range(1,n+1):
%o A333072             p = (p+ki*glist[i-1]) % f
%o A333072             ki = (k*ki) % f
%o A333072         if p == 0:
%o A333072             return k
%o A333072         k += m # _Chai Wah Wu_, Apr 04 2020
%Y A333072 Cf. A034386, A055773, A332734, A333196.
%K A333072 nonn
%O A333072 1,2
%A A333072 _Jinyuan Wang_, Mar 10 2020