[seqfan] Re: Need help with Colombian numbers etc.

Neil Sloane njasloane at gmail.com
Sat Oct 12 20:40:02 CEST 2013

```Donovan J. replied to my message:

Here are the first 16 terms > 10^13 for "Numbers m with g(m) = 3":
10000000000001, 10000000000003, 10000000000005, 10000000000007,
10000000000009, 10000000000011, 10000000000013, 10000000000015,
10000000000102, 10000000000104, 10000000000106, 10000000000108,
10000000000110, 10000000000112, 10000000000114, 10000000000116

I have checked the range [10^13, 10^13+10^11] and found no other terms.
I'll let my program run to at least 10^13+10^12 and let you know if I find
any other terms.

Regards,

Donovan

P.S. I sent this to you directly because I was not allowed to join seqfan
because I use yahoo email.

On Sat, Oct 12, 2013 at 12:37 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Consider the map f(n) = n + (sum of digits of n) (see A062028, A007953).
> Let g(m) = number of n such that f(n) = m (i.e. the number of inverses of
> m), A230093.
> Numbers m with g(m) = 0 are called the Self or Colombian numbers, A003052.
> Numbers m with g(m) = 1 give A225793.
> Numbers m with g(m) = 2 give A230094.
> The following paper:
> Narasinga Rao, A. On a technique for obtaining numbers with a multiplicity
> of generators. Math. Student 34 1966 79--84 (1967). MR0229573 (37 #5147)
> seems to say that the sequence
> Numbers m with g(m) = 3
> begins with 10^13 + 1, but I may have misread it.
> Certainly 9999999999892, 9999999999901 and 10000000000000 all have f(n) =
> 10000000000001, so that number has at least 3 inverses.
> In any case Narasinga Rao's assertion should be checked.
> Could someone compute the first few terms of "Numbers m with g(m) = 3"?
>
> In base two the analogs of these sequences are respectively
> A092391, A000120, A228085, A010061, A222088, A230091, A230092.
>
> Neil
>
>
> --
> Dear Friends, I have now retired from AT&T. New coordinates:
>
> Neil J. A. Sloane, President, OEIS Foundation
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Email: njasloane at gmail.com
>
>

--
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.