Review: A048799

[David Wilson]

A048799 needs some work.  The few values of A048799 are inaccurate, but I
cannot fix the sequence without the help of extended-precision arithmetic.
Let S(n) = A002034(n) = smallest k >= 1 with n | k!.  A048799 purports to be
the decimal expansion of the constant

  %F A048799 Sum (1/S(n)!), where S(n) is the Smarandache function A002034
and n >= 2.

For simplicity, I would like to change this formula to

  %F A048799 SUM(j = 1 to inf; 1/A002034(j)!)

which serves only to increase the constant by 1.

To compute the constant efficiently, do as follows:

Let z be the smallest number such that z! has 200+ digits (this
is probably overkill, but hey).

Now, compute a(n) = A027423(n) = d(n!) for 1 <= n <= z. I
suspect that Mma could compute d(n!) directly with fair
efficiency, if not, you can use the formula:

  a(n) = PROD(p <= n, p prime; I(n, p)+1)

where

  I(n, p) = (n-S(n, p)) / (p-1)

where

  S(n, p) = sum of base-p digits of n.

As a check, a() should start with

  1,2,4,8,16,30,60,96,160,270,...

Now, compute b(n) for 1 <= n <= z as the differences of a(n):

  b(n) = 1 if n = 1; a(n)-a(n-1) if n > 1.

b() should start with

  1,1,2,4,8,14,30,36,64,110,...

b(n) gives the number of values k with A002034(k) = n.

Finally, we compute the constant

  c = SUM(j = 1 to z; b(n)/n!).

Our choice of z should make this accurate to 108 digits.  These 108
digits constitue A048799.  As a check, you should get

  c = 2.093170459...

So we should have

  %S A048799 2,0,9,3,1,7,0,4,5,9,...

And the 108 digits should fill out the sequence.