[seqfan] Re: Chasing base-10 Harshads

Claudio Meller claudiomeller at gmail.com
Mon Sep 20 14:38:08 CEST 2010

```Hi Eric,

I found thius values :

15, 19, 14, 28, 23, 16, 22, 65, 55, 142, 134, 130, 119, 109, 95, 79, 71, 58,
47, 37, 32, 25, 17, 13, 11, 44, 256, 245, 235, 815, 1313, 1489, 1469, 1510,
1493, 1480, 1829, 1828, 1814, 1789, 1772, 3115, 4295, 4276, 4262, 4246,
4229, 4216, 4196, 4177, 4163, 4147, 4183, 4166, 4153, 4142, 4132, 4118,
4111, 4094, 4081, 8914, 8885, 8857, 8834, 8809, 8783, 8761, 8741, 8722,
8699, 8674, 8648, 8626, 8597, 8569, 8546, 8530, 8513, 8491, 8471, 8452,
8429, 8413, 8387, 8365, 8345, 8326, 8312, 8287, 8270, 8248, 8228, 8209,
8186, 8170, 8153, 8140

Best, Claudio

2010/9/20 Eric Angelini <Eric.Angelini at kntv.be>

>
> « A Harshad number, or Niven number (in a given number base),
>  is an integer that is divisible by the sum of its digits
>  (when written in that base). » [Wikipedia]
>
> Hello Seqfans,
> Let us start with 11; is 11 divisible by (1+1)=2?
> No. We then add 2 to 11 => 13
>
> Is 13 divisible by (1+3)=4?
> No. We then add 4 to 13 => 17
>
> Is 17 divisible by (1+7)=8?
> No. We then add 8 to 17 => 25
> ...
>
> Non-Harshad 11 needs 25 steps to hit 247 -- which is Harshad:
>
> 11-13-17-25-32-37-47-58-71-79-95-109-119-130-134-142-149-163-
> 173-184-197-214-221-226-236-247 (247/13=19)
>
> n   steps to reach a Harshad:
> 1     0
> 2     0
> 3     0
> 4     0
> 5     0
> 6     0
> 7     0
> 8     0
> 9     0
> 10    0
> 11    25
> 12    0
> 13    24
> 14    4
> 15    1
> 16    6
> 17    23
> 18    0
> 19    2
> 20    0
> 21    0
> ...
> 0-step are the Harshad numbers, of course:
> http://www.research.att.com/~njas/sequences/A005349<http://www.research.att.com/%7Enjas/sequences/A005349>
>
> We could build a seq where n is the required number of steps
> for the smallest a(n) to hit a Harshad; this seq would start
> like this (I think):
>
> S = 15,19,a,14,b,16,...
>
> 15 is the smallest integer needing 1 step  to hit a Harshad
> 19 is the smallest integer needing 2 steps to hit a Harshad
> a  is the smallest integer needing 3 steps to hit a Harshad
> 14 is the smallest integer needing 4 steps to hit a Harshad
> b  is the smallest integer needing 5 steps to hit a Harshad
> 16 is the smallest integer needing 6 steps to hit a Harshad
> ...
>
> Could someone compute a hundred or so terms of S (if of interest)?
> Is it possible for an integer not to hit an Harshad at some point?
>
> Best,
> É.
>
>
>
>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

--
Claudio
http://grageasdefarmacia.blogspot.com
http://todoanagramas.blogspot.com/
http://simplementenumeros.blogspot.com/

```