A332734 -id:A332734 - cp's OEIS frontend

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

Jinyuan Wang, Mar 10 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..61

Cf. A034386, A055773, A332734, A333196.

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 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

a(n) <= A034386(n).

A330030 Least k such that Sum_{i=0..n} k^n / i! is a positive integer.

Original entry on oeis.org

1, 1, 2, 3, 6, 30, 30, 42, 210, 42, 210, 2310, 2310, 30030, 30030, 30030, 30030, 39270, 510510, 1939938, 9699690, 9699690, 9699690, 17160990, 223092870, 903210, 223092870, 223092870, 223092870, 6469693230, 6469693230, 200560490130, 200560490130, 10555815270, 200560490130

Offset: 0

Jinyuan Wang, Mar 07 2020

nonn

Least k > 0 such that k^n/A061355(n) is an integer.

			For n = 7, the denominator of Sum_{i=0..7} 1/i! is 252 = 2^2*3^2*7, so a(7) = 2*3*7 = 42.

Jinyuan Wang, Table of n, a(n) for n = 0..500

Cf. A000142, A007947, A061354, A061355, A332734, A333196.

a(n) = factorback(factorint(denominator(sum(i=2, n, 1/i!)))[, 1]);

a(n) = A007947(A061355(n)).

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

Original entry on oeis.org

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

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

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A330030 Least k such that Sum_{i=0..n} k^n / i! is a positive integer.

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

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PARI

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