A328095 Revenant numbers: numbers k such that k multiplied by the product of all its digits contains k as a substring.

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

Offset: 1

Author

N. J. A. Sloane, Oct 19 2019

Keywords

• nonn
• base

Comments

Sequence is infinite since 11...1 is always a member.

Numbers whose product of digits is a power of ten (and thus necessarily must only have 1,2,4,5,8 as digits) is a subsequence. - Chai Wah Wu, Oct 19 2019

Examples

			87 * 8 * 7 = 4872. As the string 87 is visible in the result, 87 is a revenant.
So is 792 because 792 * 7 * 9 * 2 = 99792.
And so is 9375 as 9375 * 9 * 3 * 7 * 5 = 8859375.
		

References

• Eric Angelini, Posting to Sequence Fans Mailing List, Oct 19 2019

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019.
• Eric Angelini, Revenant Numbers, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]

Crossrefs

Subsequences are: A328544, A328560, A328561.

Programs

Maple

a:= proc(n) option remember; local k; if n=1 then 0 else
      for k from 1+a(n-1) while searchtext(cat(k), cat(k*
      mul(i, i=convert(k, base, 10))))=0 do od: k fi
    end:
seq(a(n), n=1..75);  # Alois P. Heinz, Oct 19 2019

Mathematica

Select[Range[0,10000],SequenceCount[IntegerDigits[#*(Times@@IntegerDigits[ #])],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 19 2019 *)

PARI

is_A328095(n)={my(d,m); if(d=vecprod(digits(n))*n, m=10^logint(n, 10)*10; until(n>d\=10,d%m==n && return(1)),!n)} \\ M. F. Hasler, Oct 20 2019

Python

from functools import reduce
from operator import mul
n, A328095_list = 0, []
while len(A328095_list) < 10000:
    sn = str(n)
    if  sn in str(n*reduce(mul,(int(d) for d in sn))):
        A328095_list.append(n)
    n += 1 # Chai Wah Wu, Oct 19 2019

Formula

A328560 union A328561.