AVirtualSpaceTimeTravelMachine : The additive persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right- (La persistance additive des 65536 premiers nombres entiers pour les bases 2 -en bas et a gauche- a 17 -en haut et a droite-)

The additive persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right- [La persistance additive 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(PA(N,B)) where:

                    PA(N,B) = the additive persistence of N for the base B,
                    f(...)  = an arbitrary ascending function

Let's define PA(N,B) with an obvious example:

                    B = 10
                                  2      1      0
                    N = 856 (= 8xB  + 5xB  + 6xB )

Then the following sequence is computed:

                    856 ---> (8+5+6) = 19 ---> (1+9) = 10 ---> (1+0) = 1
                         1                 2               3

It takes three (3) steps to reach a one digit number. Then:

                    PA(856,10) = 3

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 global renormalization- only the maximum of the sixteen maxima of PA(N,B) (B={2,3,...,16,17}) is displayed as a white point (in fact too small to be seen).

See a related picture:

See some pictures related to the multiplicative persistence:

(CMAP28 WWW site: this page was created on 05/31/2013 and last updated on 04/19/2019 14:56:36 -CEST-)

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Copyright (c) Jean-François Colonna, 2013-2019.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2019.

856 ---> (8+5+6) = 19 ---> (1+9) = 10 ---> (1+0) = 1 1 2 3

Copyright (c) Jean-François Colonna, 2013-2019. Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2019.

Copyright (c) Jean-François Colonna, 2013-2019.
Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2019.