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

0,3

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

Links

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

Formula

a(n) <= A034386(n).

Example

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.

Prog

(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

Crossrefs

Cf. A000142, A007947, A034386.

Sequence in context: A350622 A019228 A361531 * A178134 A369190 A291489

Adjacent sequences: A332731 A332732 A332733 * A332735 A332736 A332737

Keyword

nonn

Author

Jinyuan Wang, Mar 06 2020

Extensions

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

More terms from Bert Dobbelaere, Mar 09 2020

Status

approved