A332734 Least k such that Sum_{i=0..n} k^i / i! is a positive integer.

1, 1, 2, 3, 2, 30, 24, 21, 90, 126, 210, 660, 462, 8580, 6006, 1980, 4410, 157080, 39270, 2106720, 510510, 5087250, 1963500, 91861770, 29099070, 1806420, 17117100, 48498450, 135795660, 340510170, 562582020, 5642366730, 1539968430, 47683165530, 17440042620

Offset: 0

Author

Jinyuan Wang, Mar 06 2020

Keywords

• nonn

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)*e^k/n! is a positive integer, where Gamma is the upper incomplete gamma function. - Chai Wah Wu, Apr 02 2020

Examples

			For n = 4, k > 0 if Sum_{i=0..4} k^i / i! is positive. a(4) = 2 since 1 + 1/1 + 1/2 + 1/6 + 1/24 = 65/24 is not an integer and 1 + 2/1 + 4/2 + 8/6 + 16/24 = 7 is an integer.
		

Links

• Bert Dobbelaere, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000142, A007947, A034386.

Programs

PARI

a(n) = for(k=1, oo, if((s=sum(i=2, n, k^i/i!))==floor(s), return(k)));

PARI

a(n) = {if (n==0, return (1)); my(m = factorback(factorint(n)[, 1]), k = m); while (denominator(sum(i=0, n, k^i/i!)) != 1, k += m); k;} \\ Michel Marcus, Mar 06 2020

Formula

a(n) <= A034386(n).

Extensions

a(24)-a(30) from Michel Marcus, Mar 06 2020

More terms from Bert Dobbelaere, Mar 09 2020