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A333074 Least k such that Sum_{i=0..n} (-k)^i / i! is a positive integer. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

LINKS

Table of n, a(n) for n=0..35.

FORMULA

a(n) <= A034386(n).

PROG

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

(Python)

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

CROSSREFS

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

Sequence in context: A100604 A062931 A059614 * A241974 A226055 A028426

Adjacent sequences:  A333071 A333072 A333073 * A333075 A333076 A333077

KEYWORD

nonn

AUTHOR

Jinyuan Wang, Mar 31 2020

EXTENSIONS

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

STATUS

approved

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Last modified July 23 17:41 EDT 2021. Contains 346259 sequences. (Running on oeis4.)