Is a sequence involving multiplication of the previous 2 numbers' non-zero digits eventually periodic?

Asked Jan 02 '20 at 16:18

Active Jan 02 '20 at 16:18

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Assume that the first two numbers of a sequence are natural numbers $n_1$ and $n_2$ (they may be equal). Every following number in the sequence is calculated by multiplying all non-zero digits of the previous two numbers. Is this sequence eventually periodic for all $n_1$ and $n_2$?

An example with 27 and 90:$$n_1=27$$$$n_2=90$$$$n_3=2*7*9=126$$$$n_4=9*1*2*6=108$$ and so forth.

The only observation I've made is that many of these sequences appear to have a period $5040, 4200, 160, 48, 192, 576, 3780, 35280, 40320, 5760$ (not always starting with $5040, 4200$).

asked Jan 02 '20 at 16:18

haal_

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$25, 50, 50$ and repeating $1$s are also possible periods. – paw88789 Jan 02 '20 at 18:30
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http://www.codeskulptor.org/#user47_mGx4eLwMtC_0.py – Saketh Malyala Jan 02 '20 at 22:17
I've since found a couple of new periods: $n_1=5, n_2=677$, which reaches $3657830400, 3657830400, 3657830400...$, and $n_1=399, n_2=48$, which reaches $914457600, 914457600, 914457600...$. – haal_ Jan 04 '20 at 18:37

Is a sequence involving multiplication of the previous 2 numbers' non-zero digits eventually periodic?

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