# Sequence summation chains

Joshua Zucker joshua.zucker at gmail.com
Sun Jul 23 03:25:32 CEST 2006

```Hi Russell,
I think I can automate the generation of your sequences ...

I get quite a different sequence, though!
For example, I have 8, sum of prime factors 6, so 14, sum of prime
factors 9, so 23, prime, so done.
But you have a(8) = 19.

Similarly I had 9->15->23 but you have 9 ending up at 131?

For 8, I take 8, sopfr 6, sopfr 5, so 8+6+5 = 19, prime, and done.

For 9, sopfr 6, sopfr 5, and I get 20.
For 20, sopfr 9, sopfr 6, sopfr 5, and I get 40.
Continuing, 40 -> 51 -> 91 -> 131 which at last is prime.

So (starting with 1 instead of 2) I get
1 2 3 4 5 11 7 19 131 17 11 19 13 53 53 137 17 37 19 131 14419 137 23
619 61 79 47 241 29 47 31 83 67 53 137 53 37 14419 241 131 41 61 43
619 293 139 47 59 83 293 131 293 53 137 103 293 139 89 59 79 61 293
14419 83 137 101 67 233 293 14419 71 131 73 241 139 14419 317 619 79
1259 263 1399 83 179 619 971 1399 139 89 103 131 139 1259 401 139 109
97 193 14419 263

which seems to agree with you up to n = 21.  Oh, wait, it agrees with

If you want more terms or whatever, let me know!

For your other sequence, I get (again starting with n = 1)
1 1 1 1 1 2 1 2 6 2 1 2 1 3 3 6 1 2 1 5 40 6 1 9 3 3 2 7 1 2 1 3 2 2 5
2 1 39 6 4 1 2 1 8 6 4 1 2 2 5 3 5 1 4 3 5 3 2 1 2

My program is in PLT DrScheme so if Scheme/LISP is a language you
recognize I'm happy to share it as well.

--Joshua Zucker
joshua.zucker at stanfordalumni.org

On 7/20/06, Russell Walsmith <ixitol at gmail.com> wrote:
> The "sopf" (sum of prime factors) function (A001414) downgrades the
> operators in a prime decomposition. E.g., 40 factors as 2^3 * 5 and sopf(40)
> = 2 * 3 + 5 =11. Iteration of sopf gives the sequences A002217 and A029908.
> (it would be nice to have these transforms in the OEIS library BTW.)
>
> http://www.research.att.com/~njas/sequences/transforms.html
>
> An extension of this idea is to sum iterations of sopf(n) and add to n to
> generate n[2], halting finally when n[j] is prime. This gives the sequences
>
> 1, 1, 1, 1, 2, 1, 2, 6, 2, 1, 2, 1, 3, 3, 6, 1, 2, 1, 5, 31, 6
>
> 2, 3, 4, 5, 11, 7, 19, 131, 17, 11, 19, 13, 53, 53, 137, 17, 37, 19, 131,
> 5237, 137which I'll soon submit as A120978 and A120979 respectively. I have
> more terms, but it was very tedious to calculate them and I'm not certain of
> their accuracy. If anyone sees a way to automate the process, lemme know.