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A214686 Numerators of a series with denominators n! and sum 1. 2
1, 1, 7, 1, 23, 1, 47, 1, 79, 1, 113, 89, 23, 73, 31, 1, 283, 89, 113, 139, 173, 67, 47, 1, 619, 131, 109, 83, 113, 211, 191, 1, 1087, 1, 1223, 1, 1367, 1, 1511, 367, 83, 1, 1847, 1, 2017, 317, 571, 241, 199, 1, 2593, 367, 211, 271, 223, 1, 3229, 1117, 239, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

a(n) is the greatest integer x such that gcd(x,n!) = 1 and x/n! < 1 - sum_{j=2}^{n-1} a(j)/j!.

The infinite series sum_{n=2}^infinity a(n)/n! = 1

For each n, either a(n) = 1 or a(n) >= n+1.

LINKS

Robert Israel and T. D. Noe, Table of n, a(n) for n = 2..10000 (first 1000 terms from Robert Israel)

EXAMPLE

1 - a(2)/2! = 1/2 = 3/3!, gcd(2,3!)>1 so a(3) = 1.

1 - a(2)/2! - a(3)/3! = 8/4! so a(4) = 7.

MAPLE

N:= 100; a[2]:= 1; R[2]:= 1/2;

for j from 3 to N do

   T:= R[j-1] *j!;

   for x from T-1 by -1 while igcd(x, j!) > 1 do end do;

   a[j]:= x;

   R[j]:= R[j-1] - x/j!

end do:

seq(a[j], j=2..N);

MATHEMATICA

s = 1; Table[d = n! s; q = If[d <= n, 1, If[d <= n^2, Prime[PrimePi[d]], Print["d > n^2"]; Abort[]]]; s = s - q/n!; q, {n, 2, 100}] (* T. D. Noe, Jul 27 2012 *)

PROG

(Sage)

def A214686_list(n) :

    a = [1]; R = 1/2

    for j in (3..n+1) :

        J = factorial(j)

        T = R * J

        for x in range(T-1, -1, -1) :

            if gcd(x, J) == 1 : break

        a.append(x)

        R = R - x / J

    return a

A214686_list(51) # Peter Luschny, Jul 27 2012

CROSSREFS

Sequence in context: A019431 A264615 A261248 * A211790 A064051 A147385

Adjacent sequences:  A214683 A214684 A214685 * A214687 A214688 A214689

KEYWORD

nonn

AUTHOR

Robert Israel, Jul 25 2012

STATUS

approved

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Last modified September 26 10:38 EDT 2017. Contains 292518 sequences.