The multiplicative persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right- [La persistance multiplicative des 65536 premiers nombres entiers pour les bases 2 -en bas et à gauche- à 17 -en haut et à droite-].

• Starting from the origin of the coordinates (at the center of the picture), one follows a square spiral-like path and numbers each integer point encountered (1, 2, 3,...):

5----4----3
|         |    .
|         |    .
6    1----2    .
|              |
|              |
7----8----9----10

• Then one displays the N-th point with the false color f(PM(N,B)) where:
PM(N,B) = the multiplicative persistence of N for the base B,
f(...)  = an arbitrary ascending function.

Let's define PM(N,B) with an obvious example:
B = 10
1      0
N = 77 (= 7xB  + 7xB )

Then the following sequence is computed:

## 77 ---> (7x7) = 49 ---> (4x9) = 36 ---> (3x6) = 18 ---> (1x8) = 8 1 2 3 4

It takes four (4) steps to reach a one digit number (by the way it is the longest sequence with a two digit number). Then:
PM(77,10) = 4

A conjecture states that PM(N,10) cannot exceed 11...

Here is an example of a longer sequence:

## 48699984 ---> 4478976 ---> 338688 ---> 27648 ---> 2688 ---> 768 ---> 336 ---> 54 ---> 20 ---> 0 1 2 3 4 5 6 7 8 9

• More information on this subject is available in the august 2013 issue of PLS (Pour La Science) with the Jean-Paul Delahaye's paper La persistance des nombres.

In this set of 4x4 pictures -with local renormalizations- for each B (from 2 to 17) the maximum of PM(N,B) is displayed as a white point (in fact too small to be seen).

See some related pictures:

See some pictures related to the additive persistence:

(CMAP28 WWW site: this page was created on 05/31/2013 and last updated on 04/19/2019 14:56:52 -CEST-)

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