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A055471 Divisible by the product of its nonzero digits.

%I A055471 #66 Jan 20 2024 04:10:03
%S A055471 1,2,3,4,5,6,7,8,9,10,11,12,15,20,24,30,36,40,50,60,70,80,90,100,101,
%T A055471 102,104,105,110,111,112,115,120,128,132,135,140,144,150,175,200,208,
%U A055471 210,212,216,220,224,240,250,300,306,312,315,360,384,400,432,480,500
%N A055471 Divisible by the product of its nonzero digits.
%C A055471 If n is the term then 10n also is. - _Zak Seidov_, Jun 09 2013
%C A055471 De Koninck and Luca showed that the number of terms of this sequence below x is at least x^0.495 but at most x^0.901 for sufficiently large x. - _Tomohiro Yamada_, Nov 18 2017
%C A055471 This sequence begins with a run of 12 consecutive terms, from 1 to 12. The maximal length of a run of consecutive integer terms is 13. The smallest example of such a run begins with 1111011111000 and ends with 1111011111012 (Diophante link). - _Bernard Schott_, Apr 26 2019
%C A055471 These numbers are called "nombres prodigieux" on the French site Diophante. - _Bernard Schott_, Apr 26 2019
%H A055471 Marius A. Burtea, <a href="/A055471/b055471.txt">Table of n, a(n) for n = 1..11442</a> (terms 1..1000 from Zak Seidov)
%H A055471 Jean-Marie De Koninck and Florian Luca, <a href="https://doi.org/10.4171/PM/1777">Positive integers divisible by the product of their nonzero digits</a>, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)
%H A055471 Jean-Marie De Koninck and Florian Luca, <a href="https://doi.org/10.4171/PM/1999">Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85</a>, Port. Math. 74 (2017), 169-170.
%H A055471 Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a3-nombres-remarquables/3578-a365-les-nombres-prodigieux">A365, les nombres prodigieux</a>, July 2016.
%H A055471 Michael Gohn, Joshua Harrington, Sophia Lebiere, Hani Samamah, Kyla Shappell, and Tony W. H. Wong, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Wong/wong42.html">Arithmetic Progressions of b-Prodigious Numbers</a>, J. Int. Seq., Vol. 25 (2022), Article 22.8.7.
%t A055471 Select[Range[5000], IntegerQ[ #/(Times @@ Select[IntegerDigits[ # ], # > 0 &])] &] (* _Alonso del Arte_, Aug 04 2004 *)
%o A055471 (MATLAB) m=1;
%o A055471 for n=1:1000
%o A055471     v=dec2base(n,10)-'0';
%o A055471     v = v(v~=0);
%o A055471     if mod(n,prod(v))==0
%o A055471         sol(m)=n;
%o A055471         m=m+1;
%o A055471     end
%o A055471 end
%o A055471 sol % _Marius A. Burtea_, May 07 2019
%o A055471 (Magma) m:=1;sol:=[];
%o A055471 for n in [1..1000] do
%o A055471       v:=Intseq(n,10);
%o A055471        while &*v eq 0 do; Exclude(~v, 0); end while;
%o A055471      if n mod &*(v) eq 0  then ; sol[m]:=n; m:=m+1; end if;
%o A055471 end for;
%o A055471 sol // _Marius A. Burtea_, May 07 2019
%o A055471 (Python)
%o A055471 from math import prod
%o A055471 def ok(n): return n > 0 and n%prod([int(d) for d in str(n) if d!='0']) == 0
%o A055471 print(list(filter(ok, range(501)))) # _Michael S. Branicky_, Jul 27 2021
%Y A055471 Superset of A007602.
%Y A055471 Cf. A007088.
%K A055471 nonn,base
%O A055471 1,2
%A A055471 _Robert G. Wilson v_, Jul 05 2000
%E A055471 Corrected by _Patrick De Geest_, Aug 15 2000