A333074 -id:A333074 - cp's OEIS frontend

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

Jinyuan Wang, Mar 31 2020

nonn

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).

Chai Wah Wu, Table of n, a(n) for n = 1..60

Cf. A034386, A055773, A333072, A333074.

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; }

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

a(n) <= A034386(n).

cp's OEIS Frontend

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

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