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

1, 2, 6, 6, 30, 10, 70, 70, 210, 168, 1848, 1848, 18018, 8580, 2574, 2574, 102102, 102102, 831402, 2771340, 3233230, 587860, 43266496, 117630786, 162249360, 145088370, 145088370, 2897310, 672175920, 672175920, 18232771830, 18232771830, 44279588730, 8886561060

Offset

1,2

Comments

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).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..61

Formula

a(n) <= A034386(n).

Prog

(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; }
(Python)
from sympy import primorial, lcm
def A333072(n):
    f = 1
    for i in range(1, n+1):
        f = lcm(f, i)
    f, glist = int(f), []
    for i in range(1, n+1):
        glist.append(f//i)
    m = 1 if n < 2 else primorial(n, nth=False)//primorial(n//2, nth=False)
    k = m
    while True:
        p, ki = 0, k
        for i in range(1, n+1):
            p = (p+ki*glist[i-1]) % f
            ki = (k*ki) % f
        if p == 0:
            return k
        k += m # Chai Wah Wu, Apr 04 2020

Crossrefs

Cf. A034386, A055773, A332734, A333196.

Sequence in context: A144361 A366369 A163641 * A333196 A216850 A070889

Adjacent sequences: A333069 A333070 A333071 * A333073 A333074 A333075

Keyword

nonn

Author

Jinyuan Wang, Mar 10 2020

Status

approved