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A007602 Numbers that are divisible by the product of their digits.

%I A007602 M0482 #113 Mar 25 2024 06:36:22
%S A007602 1,2,3,4,5,6,7,8,9,11,12,15,24,36,111,112,115,128,132,135,144,175,212,
%T A007602 216,224,312,315,384,432,612,624,672,735,816,1111,1112,1113,1115,1116,
%U A007602 1131,1176,1184,1197,1212,1296,1311,1332,1344,1416,1575,1715,2112,2144
%N A007602 Numbers that are divisible by the product of their digits.
%C A007602 These are called Zuckerman numbers to base 10. [So-named by J. J. Tattersall, after Herbert S. Zuckerman. - _Charles R Greathouse IV_, Jun 06 2017] - Howard Berman (howard_berman(AT)hotmail.com), Nov 09 2008
%C A007602 This sequence is a subsequence of A180484; the first member of A180484 that is not a member of A007602 is 1114. - _D. S. McNeil_, Sep 09 2010
%C A007602 Complement of A188643; A188642(a(n)) = 1; A038186 is a subsequence; A168046(a(n)) = 1: subsequence of A052382. - _Reinhard Zumkeller_, Apr 07 2011
%C A007602 The terms of n digits in the sequence, for n from 1 to 14, are 9, 5, 20, 40, 117, 285, 747, 1951, 5229, 13493, 35009, 91792, 239791, 628412, 1643144, 4314987. Empirically, the counts seem to grow as 0.858*2.62326^n. - _Giovanni Resta_, Jun 25 2017
%C A007602 De Koninck and Luca showed that the number of Zuckerman numbers below x is at least x^0.122 but at most x^0.863. - _Tomohiro Yamada_, Nov 17 2017
%C A007602 The quotients obtained when Zuckerman numbers are divided by the product of their digits are in A288069. - _Bernard Schott_, Mar 28 2021
%D A007602 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007602 James J. Tattersall, Elementary Number Theory in Nine Chapters (2005), 2nd Edition, p. 86 (see problems 44-45).
%H A007602 Reinhard Zumkeller and Zak Seidov, <a href="/A007602/b007602.txt">Table of n, a(n) for n = 1..10000</a>
%H A007602 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 A007602 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 A007602 Qizheng He and Carlo Sanna, <a href="https://arxiv.org/abs/2403.14812">Counting numbers that are divisible by the product of their digits</a>, arXiv:2403.14812 [math.NT], 2024.
%H A007602 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/Zuckerman_number/">Zuckerman numbers</a>, Numbers Aplenty.
%H A007602 <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>
%p A007602 filter:= proc(n)
%p A007602 local p;
%p A007602 p:= convert(convert(n,base,10),`*`);
%p A007602 p <> 0 and n mod p = 0
%p A007602 end proc;
%p A007602 select(filter, [$1..10000]); # _Robert Israel_, Aug 24 2014
%t A007602 zuckerQ[n_] := Module[{d = IntegerDigits[n], prod}, prod = Times @@ d; prod > 0 && Mod[n, prod] == 0]; Select[Range[5000], zuckerQ] (* _Alonso del Arte_, Aug 04 2004 *)
%o A007602 (Haskell)
%o A007602 import Data.List (elemIndices)
%o A007602 a007602 n = a007602_list !! (n-1)
%o A007602 a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
%o A007602 -- _Reinhard Zumkeller_, Apr 07 2011
%o A007602 (Magma) [ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // _Bruno Berselli_, May 28 2011
%o A007602 (Python)
%o A007602 from operator import mul
%o A007602 from functools import reduce
%o A007602 A007602 = [n for n in range(1,10**5) if not (str(n).count('0') or n % reduce(mul, (int(d) for d in str(n))))] # _Chai Wah Wu_, Aug 25 2014
%o A007602 (PARI)
%o A007602 for(n=1,10^5,d=digits(n);p=prod(i=1,#d,d[i]);if(p&&n%p==0,print1(n,", "))) \\ _Derek Orr_, Aug 25 2014
%Y A007602 Cf. A034709, A001103, A005349, A288069.
%Y A007602 Cf. A286590 (for factorial-base analog).
%Y A007602 Subsequence of A002796, A034838, and A055471.
%K A007602 nonn,base,easy
%O A007602 1,2
%A A007602 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_