hakmem 56

N. J. A. Sloane njas at research.att.com
Fri Nov 20 22:55:50 CET 1998

```ITEM 56 (Beeler):

The "length" of an N-digit decimal number is defined as the number of times one mus
t iteratively form the product of its digits until
one obtains a one-digit product (see Technology Review Puzzle Corner, December 1969
and April 1970). For various N, the
following shows the maximum "length", as well as how many distinct numbers (permuta
tion groups of N digits) there are:

N     MAX L     DISTINCT
2       4           54
3       5          219
4       6          714
5       7        2,001
6       7        5,004
7       8       11,439
8       9       24,309
9       9       48,619
10      10       92,377
11      10      167.959
12      10      293,929

this is from
http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item33

The second column is clear:  the greatest mutiplicative
persistence of any n-digit number (e.g. 0, 77, 679
have m.p. 0, 4, 5)

(The persistence of a number is the number of times you need to multiply the digits together before reaching a single di
git. E.g. 39->27->14->4 has persistence 3.)

But what does the third column represent?  Neither Eric Weisstein or I can
figure it out.

NJAS

```