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

%I A328095 #58 Dec 21 2024 13:45:15
%S A328095 0,1,5,6,11,25,52,77,87,111,125,152,215,251,375,376,455,512,521,545,
%T A328095 554,736,792,1111,1125,1152,1215,1251,1455,1512,1521,1545,1554,2115,
%U A328095 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
%N A328095 Revenant numbers: numbers k such that k multiplied by the product of all its digits contains k as a substring.
%C A328095 Sequence is infinite since 11...1 is always a member.
%C A328095 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
%D A328095 Eric Angelini, Posting to Sequence Fans Mailing List, Oct 19 2019
%H A328095 Chai Wah Wu, <a href="/A328095/b328095.txt">Table of n, a(n) for n = 1..10000</a>
%H A328095 Eric Angelini, <a href="http://cinquantesignes.blogspot.com/2019/10/revenant-numbers.html">Revenant Numbers</a>, Cinquante Signes, Oct 19 2019.
%H A328095 Eric Angelini, <a href="/A328095/a328095.pdf">Revenant Numbers</a>, Cinquante Signes, Oct 19 2019. [Cached copy, pdf file, with permission]
%F A328095 A328560 union A328561.
%e A328095 87 * 8 * 7 = 4872. As the string 87 is visible in the result, 87 is a revenant.
%e A328095 So is 792 because 792 * 7 * 9 * 2 = 99792.
%e A328095 And so is 9375 as 9375 * 9 * 3 * 7 * 5 = 8859375.
%p A328095 a:= proc(n) option remember; local k; if n=1 then 0 else
%p A328095       for k from 1+a(n-1) while searchtext(cat(k), cat(k*
%p A328095       mul(i, i=convert(k, base, 10))))=0 do od: k fi
%p A328095     end:
%p A328095 seq(a(n), n=1..75);  # _Alois P. Heinz_, Oct 19 2019
%t A328095 Select[Range[0,10000],SequenceCount[IntegerDigits[#*(Times@@IntegerDigits[ #])],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 19 2019 *)
%o A328095 (Python)
%o A328095 from functools import reduce
%o A328095 from operator import mul
%o A328095 n, A328095_list = 0, []
%o A328095 while len(A328095_list) < 10000:
%o A328095     sn = str(n)
%o A328095     if  sn in str(n*reduce(mul,(int(d) for d in sn))):
%o A328095         A328095_list.append(n)
%o A328095     n += 1 # _Chai Wah Wu_, Oct 19 2019
%o A328095 (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
%Y A328095 Subsequences are: A328544, A328560, A328561.
%K A328095 nonn,base
%O A328095 1,3
%A A328095 _N. J. A. Sloane_, Oct 19 2019