# [seqfan] Re: Smallest number as sum of n palindromes

Chris Thompson cet1 at cam.ac.uk
Sat Dec 17 17:28:18 CET 2016

```Claudio Meller wrote:
> ​Hi, yesterday a friend talk to me about the paper "Every natural number is
> the sum of 49 palindromes"
> (https://arxiv.org/pdf/1508.04721.pdf)​.

Also published in INTEGERS vol 16 - http://www.westga.edu/~integers/vol16.html
item A3.

But note the rather substantial improvement by Cilleruelo & Luca in
https://arxiv.org/abs/1602.06208 !

> Based on this paper I came up with the following sequence " A(n) : Smallest
> number that can be expressed as sum of at least n different palindromes"
>
> A(1) = 0 or 1
> A(2) = 10 (9+1)
> A(3) = 21 (11+9+1)
>
> More terms?
> Is it interesting?

On Dec 17 2016, David Wilson wrote:

>Actually, the smallest number that can be expressed as sum of at least n
>different palindromes is n.

Surely, "the sum of the first n palindromes" if they have to be distinct.

>I think you want the smallest number that cannot be expressed as the sum
>of fewer than n (distinct) palindromes.

I think the Cilleruelo & Luca result can be fairly easily adapted to show
that all integers are the sum of no more than three *distinct* palindromes.

--
Chris Thompson
Email: cet1 at cam.ac.uk

```