A214686 Numerators of a series with denominators n! and sum 1.

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

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