A333072 - OEIS

%I #15 Apr 04 2020 16:29:43

%S 1,2,6,6,30,10,70,70,210,168,1848,1848,18018,8580,2574,2574,102102,

%T 102102,831402,2771340,3233230,587860,43266496,117630786,162249360,

%U 145088370,145088370,2897310,672175920,672175920,18232771830,18232771830,44279588730,8886561060

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

%C 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 Chai Wah Wu, <a href="/A333072/b333072.txt">Table of n, a(n) for n = 1..61</a>

%F a(n) <= A034386(n).

%o (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 (Python)

%o from sympy import primorial, lcm

%o def A333072(n):

%o     f = 1

%o     for i in range(1,n+1):

%o         f = lcm(f,i)

%o     f, glist = int(f), []

%o     for i in range(1,n+1):

%o         glist.append(f//i)

%o     m = 1 if n < 2 else primorial(n,nth=False)//primorial(n//2,nth=False)

%o     k = m

%o     while True:

%o         p,ki = 0, k

%o         for i in range(1,n+1):

%o             p = (p+ki*glist[i-1]) % f

%o             ki = (k*ki) % f

%o         if p == 0:

%o             return k

%o         k += m # _Chai Wah Wu_, Apr 04 2020

%Y Cf. A034386, A055773, A332734, A333196.

%K nonn

%O 1,2

%A _Jinyuan Wang_, Mar 10 2020