# [seqfan] Working in base 2, 3, k, replace n by the concatenation of its prime divisors in decreasing/increasing order (write answer in base 10) with or without multiplicity

Jonathan Post jvospost3 at gmail.com
Wed Jun 2 11:05:11 CEST 2010

```Do we have the analogue of A048985  Working in base 2, replace n by
the concatenation of its prime divisors in increasing order (write
with "decreasing" substituted for "increasing"?

n  a(n)
1   1
2   2
3   3
4   10
5   5
6   14  because 6 = 3*2 = (11 base 2)*(10 base 2) -> 1110 base 2 -> 14 base 10
7   7
8   42 because 8 = 2*2*2 -> concatenate (10, 10, 10) = 101010 base 2
-> 42 base 10
9   15
10  22
11  11
12  58 because 3*2*2 -> concatenate(11,10,10) -> 111010 base 2 -> 58 base 10

Note that A048985 seems NOT to mean "unique prime divisors" as one
gets a different result for "prime divisors with multiplicity"
starting with a(4) being (with multiplicity) Concatenate(10, 10)  =
1010 which, base 2, is 10.  If we take unique prime divisors in the
definition, we can get different results for squareful numbers.
a(4) = 10 base 2 which becomes 2 base 10.

We still have a(p=prime) = p.  But now, with "unique prime factor" in
definition we also have:
a(p^n) = p.

I'm getting a "Service Temporarily Unavailable
The server is temporarily unable to service your request due to
maintenance downtime or capacity problems. Please try again later."

so I can't check further, and am about to go to sleep.

Oh.  That message vanished, and indeed this is actually
A112417 Sequence is generated as in A048985, but primes are instead
written in decreasing order before being concatenated.

Still, my query about "unique prime divisors" in the definition stands.

Base 3 we have the analogue:
Working in base 3, replace n by the concatenation of its prime
divisors in increasing order (write answer in base 10).
This can be read off from
A001363  	 	 Primes in ternary.
giving:
n   a(n)
1   1
2   2
3   3
4   8 because 4 = 2*2 -> (2 base 3)*(2 base 3) -> Concatenate (2,2) ->
22 base 3 -> 8 base 10
5   5
6   21  because 6 = 2*3 -> (2 base 3)*(10 base 3) -> 210 base 3 -> 21 base 10
7   7
8  26
9  30
10 23 because 2,12 -> 212 base 3 -> 23 base 10
11 11
12 75
13 13
14 25
15 32
16 80
17 80
18 192 because 2*3*3 -> 2,10,10 -> 21010 base 3 = 192 base 10
19 19
20 77
21 34
22 65
23 23
24 237
25 50
26 67
27 273
28 79
29 29
30 194
modulo errors I've made in base conversion
which would be new to OEIS.

Again, in such base seqs I like to give a comment that shows the more
general case of the array A[k,n] = the n-th value of Working in base
k, replace n by the concatenation of its prime divisors in increasing
order (write answer in base 10). And then note that the 2nd row of
this array A[2,n] = A048985, and the 3rd row A[3,n] is the one I'vce
just tabulated above, and dhow the start for k = 4,5,6,7 up to perhaps
10, which is already in OEIS.

```