Smallest n-Smith numbers

[David Wilson]

Observing the moratorium on OIES submissions, I hereby post the following to 
seqfan for comment and later processing.

a(n) = smallest k >= 1 with (sum of digits of prime factors of k with 
multiplicity)/(sum of digits of k) = n.
(0 <= n <= 45).

1 2 32 402 2401 2030 10112 10 200 10200 10010 100200 1000110 1000200 100
20000 10200000 1001000 100200000 1000110000 1000200000 1000 2000000
10200000000 100100000 100200000000 1000110000000 1000000000100 10000
200000000 10200000000000 10010000000 100200000000000 1000110000000000
100000000010000 100000 20000000000 1000000000000010 1001000000000
100200000000000000 1000110000000000000 10000000001000000 1000000
2000000000000 10000000000000001 100100000000000

Replacing a(1) = 2 with a(1) = 4 gives the smallest n-Smith numbers as 
defined on Mathworld's "Smith Number" page.  This is necessary because Smith 
number omit the primes.

An easy bound a(n) <= 7*10^(k-1).  A tighter but more complicated bound 
gotten from the computed values is:

a(7k) = 10^k
a(7k+1) <= 2*10^(2k)
a(7k+2) <= 102*10^(3k-1)
a(7k+3) <= 1001*10^(2k-1)
a(7k+4) <= 1002*10^(3k-1)
a(7k+5) <= 100011*10^(3k-2)
a(7k+6) <= 10002*10^(3k-1)

for k >= 1.

I highly suspect that

lim n->inf (a(7+n)/a(n)) = 10 if n = 0, 100 if 1 <= n <= 6.

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- David Wilson