Multiplicative persistence computation

Multiplicative persistence of an integer is the number of steps to reach a one-digit number by repeated multiplication of its own digits.
Ex: N=777 → 7*7*7=343 → 3*4*3=36 → 3*6=18 → 1*8=8 : p(777)=4
"Erdős" variation consider 0's as 1's. We'll write it pE(N) instead of p(N). Ex: N=1084 → 1*1*8*4=32 → 3*2=6 : pE(1084)=2 (while p(1084)=1)

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Find more information in this excellent 2013 paper on persistence by J.-P. Delahaye, also in Pour la Science #430 paper.
This video (in french) is very clear : La persistance des nombres - Conjecture #1, Yvan Monka.
You can also check other results on Other information on, and Sloane's sequence A014120.

This page lets you compute multiplicative persistence (and Erdős variation) of any number.
It also presents new results on persistence of specific n-digits numbers like 2222...2222, 333...333, etc more generally noted x(n).

N= - Add to N:
Try specific samples :


draw f(x)=p(x) for x=[1,]
draw f(x)=pErdős(x) for x=[i,ii,iii,iiii,...] i= (x=)
draw y=f(x) where x=pE and y=log(jmin) / pE(i(j))=x. Ex : i=2, pE=18, pE(2(j))=18 => jmin=9308 (draw)
draw f(x)=freq(x) for x=[1,]
draw f(x,y)=p(xy) for x=[1,1000],y=[1,200]
For any question/suggestion :