Need help with 2421 sequence

[Tanya Khovanova]

2421 is the unique number with the property you describe.

Let f(n) = SUM(digits d of n; prime(d)^d), so that f(2421) = 2421, as you 
describe.

Let n have d digits. An inductive proof shows that for d >= 15

    f(n) <= d*prime(9)^9 = d*23^9 < 10*(d-1) <= n

meaning that for numbers of 15 or more digits, we can never have n = f(n), 
the sort of number Tanya is seeking.

This means we only have to search numbers <= 14 digits, still a daunting 
computation on an average PC.

We can of course eliminate numbers that have the digit 0, which mitigates 
the computation somewhat.

But now suppose n is one of the numbers that Tanya is looking for, that is, 
f(n) = n. If we sort the digits of n to get m, then f(m) = n and f(f(m)) = 
n. So every number n Tanya wants shows up as n = f(m) = f(f(m)) for some 
sorted zeroless m of 14 or fewer digits, a MUCH smaller search space.

I wrote a program based on these observations that confirmed that 2421 is 
the unique solution in less than 2 seconds.

----- Original Message ----- 
From: "Tanya Khovanova" <tanyakh at TanyaKhovanova.com>
To: <seqfan at ext.jussieu.fr>
Sent: Friday, September 14, 2007 1:33 PM
Subject: Need help with 2421 sequence


> Hello seqfans,
>
> I am trying to build a sequence corresponding to the following property of 
> 2421:
>
> 2421 = prime(2)^2 + prime(4)^4 + prime(2)^2 + prime(1)^1.
>
> Comment: this property is not defined for numbers with zeroes as their 
> digits.
>
> My Mathematica coding is too slow,  I can only check up to 100000000.
>
> This sequence is obviously finite. I know theoretical ways to speed up my 
> code, but practically I am not that good yet :-)
>
> If there are no other numbers like this, it would be very interesting for 
> me personally, as it would mean that 2421 is the ONLY number with this 
> property. In my number gossip database there are about 1000 unique 
> properties and only a very small proportion correspond to "the only" 
> property (as opposed to "the smallest" property).
>
> Uniqueness of this property would mean a great deal to me especially as 
> this number is relatively big. If there are other numbers like this, there 
> would be a new sequence. So we win in any case :-)
>
> Best, Tanya
>
>
> _________________________________________________________________
> Need personalized email and website? Look no further. It's easy
> with Doteasy $0 Web Hosting! Learn more at www.doteasy.com
>
>
> -- 
> No virus found in this incoming message.
> Checked by AVG Free Edition.
> Version: 7.5.487 / Virus Database: 269.13.19/1008 - Release Date: 
> 9/14/2007 8:59 AM
>