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A332734 Least k such that Sum_{i=0..n} k^i / i! is a positive integer. 4
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 (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)*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: A025522 A350622 A019228 * A178134 A291489 A075121

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

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Last modified August 10 12:36 EDT 2022. Contains 356039 sequences. (Running on oeis4.)