A333073 -id:A333073 - cp's OEIS frontend

A333074 Least k such that Sum_{i=0..n} (-k)^i / i! is a positive integer.

1, 1, 2, 3, 4, 30, 6, 28, 120, 84, 210, 1650, 210, 11440, 6930, 630, 9240, 353430, 93450, 746130, 1616160, 746130, 1021020, 11104170, 56705880, 9722790, 48498450, 174594420, 87297210, 222071850, 2114532420, 11480905800, 5375910540, 42223261080, 5603554110, 2043061020

Offset: 0

Jinyuan Wang, Mar 31 2020

nonn

Note that Sum_{i=0..n-1} (-k)^i / i! has a denominator that divides (n-1)! for n > 0. Therefore, for the expression to be an integer, (-k)^n / n! must have a denominator that divides (n-1)!. Thus, k^n is divisible by n, a(n) = k is divisible by A007947(n).

a(n) is the smallest integer k such that Gamma(n+1,-k)/(n!*e^k) is a positive integer, where Gamma is the upper incomplete gamma function. - Chai Wah Wu, Apr 01 2020

Cf. A000142, A007949, A034386, A332734, A333073.

a(n) = {my(m = factorback(factorint(n)[, 1]), k = m); while(denominator(sum(i=2, n, (-k)^i/i!)) != 1, k += m); !n+k; }

from functools import reduce
from operator import mul
from sympy import primefactors, factorial
def A333074(n):
    f, g = int(factorial(n)), []
    for i in range(n+1):
        g.append(int(f//factorial(i)))
    m = 1 if n < 2 else reduce(mul, primefactors(n))
    k = m
    while True:
        p, ki = 0, 1
        for i in range(n+1):
            p = (p+ki*g[i]) % f
            ki = (-k*ki) % f
        if p == 0:
            return k
        k += m # Chai Wah Wu, Apr 01 2020

a(n) <= A034386(n).

a(27)-a(35) from Chai Wah Wu, Apr 01 2020

cp's OEIS Frontend

A333074 Least k such that Sum_{i=0..n} (-k)^i / i! is a positive integer.

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