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User:Daniel Mondot/Multiplicative Persistence

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Current work on multiplicative persistence

multiplicative persistence tree

The file File:Multiplicative Persistence Tree.txt gives the path of (almost) all numbers back to a single digit. Numbers that are not of the path of another number are not part of this table.
Also, because there are an infinite number of numbers that are part of a path, and lead to 0 in 1 step. Only the ones that have an ascendant also in this table are listed.

Number in this file are located according to these rules:

1st column: multiplicative persistence of 0
2nd column: multiplicative persistence of 1
3rd column: multiplicative persistence of 2
4th column: multiplicative persistence of 3
5th column: multiplicative persistence of 4
6th column: multiplicative persistence of 5
7th column: multiplicative persistence of 6
8th column: multiplicative persistence of 7
9th column: multiplicative persistence of 8
10th column: multiplicative persistence of 9
11th column: multiplicative persistence of 10

Applying the multiplicative persistence process , the product of the digits of a number p(n) is the first number that is located up and to the left of the original number.
For example, p(117649) is 1512, which is the first number found to the left and up from 117649.

Numbers of 7-smooth numbers with multiplicative persistence 1 through 11.
multiplicative persistence
ending in 0 1 2 3 4 5 6 7 8 9 10 11
0 1 11962 330 82 24 8 7 5 2 2 0
1 1 0
2 1 3 7 12 9 1 0
3 1 0
4 1 1 2 4 1 0
5 1 1 3 4 3 0
6 1 3 7 18 36 14 4 1 0
7 1 0
8 1 5 13 19 11 2 0
9 1 0

Please note that in A003001, A.H.M. Smeets has a similar table in the comment section. His numbers are incorrect, except for final digits 1,3,5,7 and 9.

From a comment A003001 by Benjamin Chaffin in 2016, there are no more numbers with multiplicative persistence >1, between 10^140 and 10^20000.
I have been running my own search up to 10^30000, and finally, after 85.6 days, the program finished and didn't find any more numbers.

Some sequences about Multiplicative Persistence

Multiplicative persistence 1 to 9:
sequence offset data Programs comment
A046510: 1,1 10000 mma/pari/python some
A046511: 1,1 10000 mma/python none
A046512: 1,1 10000 mma/maple none
A046513: 1,1 10000 mma/maple none
A046514: 1,1 10000 maple none
A046515: 1,1 10000 maple none
A046516: 1,1 10000 maple none
A046517: 1,1 10000 mma/maple none
A046518: 1,1 10000 maple none
A352531: 1,1 (10000) (done in C) none in progress
A352532: 1,1 (10000) (done in C) none in progress


Multiplicative persistence 1 to 9 and prime:
sequence offset data Programs comments
A046501: 1,1 10000 mma/pari/python some
A046502: 1,1 10000 ----------- none
A046503: 1,1 10000 mma/maple none
A046504: 1,1 10000 mathematica none
A046505: 1,1 10000 mathematica none
A046506: 1,1 10000 maple none
A046507: 1,1 10000 ----------- none
A046508: 1,1 10000 mathematica none
A046509: 1,1 10000 ----------- none
multiplicative persistence 10 and 11 might be missing.


Multiplicative persistence 1 through 10 and 7-smooth.
sequence offset data Programs comments
A350180: 1,1 20000 pari yes
A350181: 1,1 11994 mms/python yes
A350182: 1,1 387 ---------- yes
A350183: 1,1 142 mma/python/pari yes
A350184: 1,1 41 mma/python yes
A350185: 1,1 12 mma/python yes
A350186: 1,1 8 mathematica yes
A350187: 1,1 5 ----------- yes
A350188: Sequence was recycled for being too short. It was supposed to contain: 438939648, 231928233984
A350189: Sequence was recycled for being too short. It was supposed to contain: 4996238671872, 937638166841712

Biggest issues with the sequences above

- Some sequences do not have programs
- comments and perhaps some crossrefs are lacking.
- need to find a place for defunct A350188 & A350189 data