A286590 -id:A286590 - cp's OEIS frontend

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

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

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

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

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).

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

Cf. A034709, A001103, A005349, A288069.
Cf. A286590 (for factorial-base analog).
Subsequence of A002796, A034838, and A055471.

import Data.List (elemIndices)
a007602 n = a007602_list !! (n-1)
a007602_list = map succ $ elemIndices 1 $ map a188642 [1..]
-- Reinhard Zumkeller, Apr 07 2011

[ n: n in [1..2144] | not IsZero(&*Intseq(n)) and IsZero(n mod &*Intseq(n)) ];  // Bruno Berselli, May 28 2011

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

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 *)

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

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

A118363 Factorial base Niven (or Harshad) numbers: numbers that are divisible by the sum of their factorial base digits.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 26, 27, 30, 35, 36, 40, 48, 52, 54, 56, 60, 70, 72, 75, 80, 90, 91, 96, 105, 108, 112, 117, 120, 122, 123, 126, 132, 135, 140, 144, 148, 150, 152, 156, 161, 168, 175, 180, 186, 192, 204, 208, 210, 222, 224, 240, 244, 245, 246

Offset: 1

Alonso del Arte, May 15 2006

Also called "Fiven" numbers [Dahlenberg and Edgar]. - N. J. A. Sloane, Jun 25 2018

			a(8) = 16 because it is written 220 in factorial base and 2 + 2 + 0 = 4, which is a divisor of 16.
17 is not on the list because it is written 221 in factorial base and 2 + 2 + 1 = 5, which is not a divisor of 17.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Paul Dahlenberg and Tom Edgar, Consecutive factorial base Niven numbers, The Fibonacci Quarterly, Vol. 56, No. 2 (2018), pp. 163-166.
Index entries for sequences related to factorial base representation.

Cf. A007623 (integers written in factorial base), A005349 (base 10 Harshad numbers).
Cf. A286607 (complement), A034968, A286590.
Positions of zeros in A286604.

(*For the definition of the factorial base version of IntegerDigits, see A007623*) Select[Range[250],IntegerQ[ #/(Plus@@factBaseIntDs[ # ])]&]

is(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); !(n % s);} \\ Amiram Eldar, Oct 08 2024

def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

A208575 Product of digits of n in factorial base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 6, 0, 12, 0, 0, 0, 9, 0, 18

Offset: 0

Charles R Greathouse IV, Feb 28 2012

Antti Karttunen, Table of n, a(n) for n = 0..10080
M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdős concerning the persistence of numbers, Journal of Recreational Mathematics 29:2 (1998), pp. 89-92.
Index entries for sequences related to factorial base representation

Cf. A007623, A007954, A031346, A208576, A208277, A227153, A227154, A286590.

(* For the definition of the factorial base version of IntegerDigits, see A007623 *) Table[Times@@factBaseIntDs[n], {n, 0, 99}] (* Alonso del Arte, Feb 28 2012 *)

a(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s

from functools import reduce
from operator import mul
def A(n, p=2):
    return n if nIndranil Ghosh, Jun 19 2017

A341433 Numbers that are divisible by the product of their digits in primorial base representation.

Original entry on oeis.org

1, 3, 9, 21, 39, 51, 99, 249, 261, 309, 669, 729, 2559, 2571, 2619, 2979, 3051, 4239, 7179, 7191, 32589, 32601, 32649, 32661, 33009, 33021, 37209, 37269, 37629, 51489, 92649, 92709, 93069, 97281, 272889, 274509, 543099, 543111, 543159, 543519, 543591, 544779

Offset: 1

Amiram Eldar, Feb 11 2021

The primorial base repunits (A143293) are all terms since their product of digits in primorial base is 1.

All the terms are zeroless in primorial base, and therefore they are terms of A328574. In particular, since the last digit of even numbers in primorial base is 0, all the terms are odd numbers.

			9 is a term since 9 in primorial base is 111 (9 = 3! + 2! + 1!) and 9 is divisible by 1*1*1 = 1.

Amiram Eldar, Table of n, a(n) for n = 1..150
Wikipedia, Primorial number system.
Index entries for sequences related to primorial base.

A143293 is a subsequence.
Subsequence of A328574.
Cf. A007602, A049345, A235168, A286590, A333426.

max = 12; bases = Prime@Range[max, 1, -1]; nmax = Times @@ bases - 1; q[n_] := FreeQ[(d = IntegerDigits[n, MixedRadix[bases]]), 0] && Divisible[n, Times @@ d]; Select[Range[1, 10^5, 2], q]

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

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A118363 Factorial base Niven (or Harshad) numbers: numbers that are divisible by the sum of their factorial base digits.

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A208575 Product of digits of n in factorial base.

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A341433 Numbers that are divisible by the product of their digits in primorial base representation.

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Mathematica