Revenant numbers



Take an integer abc...z and multiply it by all its digits: if the string abc...z appears in the result, we have a "revenant number".

Look at 87 for instance: 87 * 8 * 7 = 4872. As the string 87 is visible in the result, 87 is a revenant.
The same with 792 because 792 * 7 * 9 * 2 = 99792.
And with 9375 as 9375 * 9 * 3 * 7 * 5 = 8859375.

If we start a sequence R of such revenants we get:

R = 0, 1, 5, 6, 11, 25, 52, 77, 87, 111, 125, 152, 215, 251, 375, 376, 455, 512, 521, 545, 554, 736, 792, 1111, 1125, 1152, 1215, 1251, 1455, 1512, 1521, 1545, 1554, 2115, 2151, 2174, 2255, 2511, 2525, 2552, 4155, 4515, 4551, 5112, 5121, 5145, 5154, 5211, 5225, 5252, 5415, 5451, 5514, 5522, 5541, 5558, 5585, 5855, 8555, 8772, 9375,...

R is infinite, of course, as all repunits (like 11, 111, 1111, 1111,...) will be in R.


Jean-Marc Falcoz has computed all revenants < 100 000 000 (they are 7607) and a few interesting things appear:

Digit-frequency in R {digit followed by its quantity in the revenants < 100 000 000}:

{0,1},{1,16100},{2,9283},{3,5},{4,4800},{5,25434},{6,6},{7,12},{8,2191},{9,5}.

[There are only 4 revenants < 100 000 000 showing at least a 9: {792, 9375, 9859155, 62227496} and there are only 5 such revenants with a 3: {375, 376, 736, 9375, 23255814}.]

Revenants < 100 000 000 whose "image" doesn't include one or more zeros {revenant followed by its image}:

{1,1},{5,25},{6,36},{11,11}, {77,3773},{87,4872},{111,111},{375,39375},{376,47376},{736, 92736},{792,99792},{1111,1111},{2174,121744},{8772,6877248},{9375,8859375},{11111,11111},{11628,1116288},{111111,111111},{1111111,1111111},{11111111, 11111111},{62227496, 4516222749696}.

[The revenant in yellow is a gem: will someone find a bigger such one?]

The last ten revenants < 100 000 000 are:
85555514, 85555522, 85555541, 85555558, 85555585, 85555855, 85558555, 85585555, 85855555, 88555555.

Do you see a pattern there?-)
Best,
É.
(the sequence is now in the OEIS – and has inspired this one)
____________________
Update (october 21st, 2019)
Chai Wah Wu (yesterday): « The next term after 62227496 that doesn't contain a 0 in the product is 6886826188 with the product being 97488368868261888 ».
Bravo Chai Wah and many thanks!


... Mais le "product" cité par Chai Wah peut très bien comporter un zéro interne ! Jean-Marc m'envoya d'ailleurs ceci (qui vient avant le terme de Chai Wah) :

Après 24h non stop de pêche au gros, j'ai fini par en ferrer un  :o) 
3'691'262'781 qui donne 803691262781568
> À première vue, un nombre de cette taille à environ une chance sur 2 milliards de marcher... 

Nombre que confirma Hans Haverman sur le forum SeqFans :

>> CWW: "The next term after 62227496 that doesn't contain a 0 in the product is 6886826188 with the product being 97488368868261888."

Hans Haverman:
> When Eric suggested that his "gems" subsequence have images/products not containing any zeros, I thought he was being unduly restrictive. Ignoring products whose *final digit* is zero reproduces his list and catches additional gems such as 3691262781 -> 803691262781568.
____________________
Update again, (october 24th 2019):
Giovanni Resta has sent this message to the SeqFans list:

I added 3 further terms to the sequence517322161894, 774773248793 and 2675959368829.

517322161894 * 725760 = 375451732216189440
774773248793 * 348509952 = 270016187747732487936
2675959368829 * 3527193600 = 9438626759593688294400


Still searching, but no other up to 2.5*10^12.
...
____________________
Merci, thanks to all and bravo!
É.















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