

A007602


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


75



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
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OFFSET

1,2


COMMENTS

These are called Zuckerman numbers to base 10. [Sonamed 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
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 4445).


LINKS



MAPLE

filter:= proc(n)
local p;
p:= convert(convert(n, base, 10), `*`);
p <> 0 and n mod p = 0
end proc;


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 !! (n1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
(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. A286590 (for factorialbase analog).


KEYWORD

nonn,base,easy


AUTHOR



STATUS

approved



