[seqfan] A computational challenge from 1973

[Neil Sloane]

Dear Seq Fans, This Spring I started going through
my file of issues of the newsletter Popular Computing.
I am now back to 1973.

In Popular Computing, Vol. 1, Number 8, November 1973,
page PC8-14, Problem 22 asks for the computation
of various decimal numbers arising from sequences.

Suppose m is a k-digit number. Let c(m) = m/100^k.
Thus, if m is a 2-digit number, c(m)  = .00m
Example: m=16, c(m)=.0016 = 16/100^2.
If m=127, c(m)=.000127 = 127/10^6.

HERE IS THE PROBLEM:
For a sequence a = a(1), a(2), a(3), ...,
define f(a) = Sum_{n >= 1} c(a(n)).

For example, take a = 1 4 9 16 25 36 ... the nonzero squares
Then f(a) is the infinite sum of
.01
.04
.09
.0016
.0025
.0036
...

According to Popular Computing, my old friend Hermann P. Robinson
computed that f(a) = .18190589020080121567...
Problem 22 asks for more digits, and I'm asking for someone
to enter this sequence (the sequence of decimal digits,
with keyword cons, as usual).

Popular Computing and I are also interested in the numbers that arise
from the cubes, powers of 2, powers of 3, Fibonacci numbers,
factorials, and subfactorials.

And (this is me) the Catalan numbers, the Motzkin numbers.

If a = primes, f(a)=oo, so we don't consider that one!

Popular Computing gives H P Robinson's values
for many of these, to 20 decimal
places, so that will be a good check on your work!