[seqfan] Chasing base-10 Harshads
Eric Angelini
Eric.Angelini at kntv.be
Mon Sep 20 13:23:24 CEST 2010
« 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
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.
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