A333073 Least k such that Sum_{i=1..n} (-k)^i / i is a positive integer.

1, 2, 6, 6, 30, 20, 140, 140, 210, 42, 462, 462, 12012, 3432, 6006, 6006, 87516, 87516, 1108536, 3048474, 2586584, 2586584, 44618574, 44618574, 60843510, 17160990, 17160990, 14263680, 782050830, 782050830, 3806842470, 3806842470, 16830250920, 16830250920

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..60

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 A333073(n):
    f = 1
    for i in range(1, n+1):
        f = lcm(f, i)
    f = int(f)
    glist = []
    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 02 2020

Crossrefs

Cf. A034386, A055773, A333072, A333074.

Sequence in context: A072744 A056042 A056040 * A099566 A147299 A090549

Adjacent sequences: A333070 A333071 A333072 * A333074 A333075 A333076

Keyword

nonn

Author

Jinyuan Wang, Mar 31 2020

Status

approved