A007602 Numbers that are divisible by the product of their digits. (Formerly M0482)

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384, 432, 612, 624, 672, 735, 816, 1111, 1112, 1113, 1115, 1116, 1131, 1176, 1184, 1197, 1212, 1296, 1311, 1332, 1344, 1416, 1575, 1715, 2112, 2144

Offset

1,2

Comments

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

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

Complement of A188643; A188642(a(n)) = 1; A038186 is a subsequence; A168046(a(n)) = 1: subsequence of A052382. - Reinhard Zumkeller, Apr 07 2011

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

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

The quotients obtained when Zuckerman numbers are divided by the product of their digits are in A288069. - Bernard Schott, Mar 28 2021

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

James J. Tattersall, Elementary Number Theory in Nine Chapters (2005), 2nd Edition, p. 86 (see problems 44-45).

Links

Reinhard Zumkeller and Zak Seidov, Table of n, a(n) for n = 1..10000

Jean-Marie De Koninck and Florian Luca, Positive integers divisible by the product of their nonzero digits, Port. Math. 64 (2007) 75-85. (This proof for upper bounds contains an error. See the paper below)

Jean-Marie De Koninck and Florian Luca, Corrigendum to "Positive integers divisible by the product of their nonzero digits", Portugaliae Math. 64 (2007), 1: 75-85, Port. Math. 74 (2017), 169-170.

Qizheng He and Carlo Sanna, Counting numbers that are divisible by the product of their digits, arXiv:2403.14812 [math.NT], 2024.

Giovanni Resta, Zuckerman numbers, Numbers Aplenty.

Index entries for Colombian or self numbers and related sequences

Maple

filter:= proc(n)
local p;
p:= convert(convert(n, base, 10), `*`);
p <> 0 and n mod p = 0
end proc;
select(filter, [$1..10000]); # Robert Israel, Aug 24 2014

Mathematica

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 *)

Prog

(Haskell)
import Data.List (elemIndices)
a007602 n = a007602_list !! (n-1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
-- Reinhard Zumkeller, Apr 07 2011
(Magma) [ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // Bruno Berselli, May 28 2011
(Python)
from operator import mul
from functools import reduce
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
(PARI)
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

Crossrefs

Cf. A034709, A001103, A005349, A288069.

Cf. A286590 (for factorial-base analog).

Subsequence of A002796, A034838, and A055471.

Sequence in context: A308472 A064700 A180484 * A343681 A337941 A167620

Adjacent sequences: A007599 A007600 A007601 * A007603 A007604 A007605

Keyword

nonn,base,easy

Author

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

Status

approved