A002796 -id:A002796 - cp's OEIS frontend

A342855 Numbers that contain a digit 0 (A011540) and that are divisible by their nonzero digits (A002796).

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 104, 105, 110, 120, 140, 150, 200, 202, 204, 208, 210, 220, 240, 250, 280, 300, 303, 306, 330, 360, 400, 404, 408, 420, 440, 480, 500, 505, 510, 520, 540, 550, 600, 606, 630, 660, 700, 707, 770, 800, 808, 840, 880, 900, 909, 990, 1000

Offset: 1

Bernard Schott, Mar 25 2021

If m is a term, so is 10*m.

			208 = 2*104 = 8*26 is divisible by 2 and divisible by 8, 208 contains a digit 0, hence 208 is a term.

Index entries for 10-automatic sequences.

Intersection of A002796 and A011540.

q[n_] := MemberQ[(d = IntegerDigits[n]), 0] && AllTrue[d, # == 0 || Divisible[n, #] &]; Select[Range[1000], q] (* Amiram Eldar, Mar 25 2021 *)

isok(m) = my(d=digits(m)); if (vecmin(d), return (0)); d = vecsort(select(x->(x > 0), d),,8); for (k=1, #d, if (m % d[k], return(0))); return(1); \\ Michel Marcus, Mar 29 2021

a(n) ~ 2520n. - Charles R Greathouse IV, Nov 25 2024

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

Original entry on oeis.org

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

A034838 Numbers k that are divisible by every digit of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128, 132, 135, 144, 155, 162, 168, 175, 184, 212, 216, 222, 224, 244, 248, 264, 288, 312, 315, 324, 333, 336, 366, 384, 396, 412, 424, 432, 444, 448

Offset: 1

Erich Friedman

Subset of zeroless numbers A052382: Integers with at least one digit 0 (A011540) are excluded.
A128635(a(n)) = n.
Contains in particular all repdigits A010785 \ {0}. - M. F. Hasler, Jan 05 2020
The greatest term such that the digits are all different is the greatest Lynch-Bell number 9867312 = A115569(548) = A113028(10) [see Diophante link]. - Bernard Schott, Mar 18 2021
Named "nude numbers" by Katagiri (1982-83). - Amiram Eldar, Jun 26 2021

			36 is in the sequence because it is divisible by both 3 and 6.
48 is included because both 4 and 8 divide 48.
64 is not included because even though 4 divides 64, 6 does not.

Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227. See problem number 2693.
Yoshinao Katagiri, Letter to the editor of the Journal of Recreational Mathematics, Vol. 15, No. 4 (1982-83).
Margaret J. Kenney and Stanley J. Bezuszka, Number Treasury 3: Investigations, Facts And Conjectures About More Than 100 Number Families, World Scientific, 2015, p. 175.
Thomas Koshy, Elementary Number Theory with Applications, Elsevier, 2007, p. 79.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Diophante, A1916. Le plus grand entier divisible par ses propres chiffres (in French).
Giovanni Resta, nude numbers, Numbersaplenty, 2013.
Roberto A. Ribas, The Nude Numbers, The Pentagon, Vol. 45, No. 1 (1985), pp. 18-31.
Voodooguru, Nude Numbers, Mathematical Meanderings, Oct 11 2020.
Eric Weisstein's World of Mathematics, Digit
Index entries for 10-automatic sequences.

Intersection of A002796 (numbers divisible by each nonzero digit) and A052382 (zeroless numbers), or A002796 \ A011540 (numbers with digit 0).
Subsequence of A034709 (divisible by last digit).
Contains A007602 (multiples of the product of their digits) and subset A059405 (n is the product of its digits raised to positive powers), A225299 (divisible by square of each digit), and A066484 (n and its rotations are divisible by each digit).
Cf. A113028, A346267 (number of terms with n digits), A087140 (complement).
Supersequence of A115569 (with all different digits).

a034838 n = a034838_list !! (n-1)
a034838_list = filter f a052382_list where
   f u = g u where
     g v = v == 0 || mod u d == 0 && g v' where (v',d) = divMod v 10
-- Reinhard Zumkeller, Jun 15 2012, Dec 21 2011

[n:n in [1..500]| not 0 in Intseq(n) and #[c:c in [1..#Intseq(n)]| n mod Intseq(n)[c] eq 0] eq #Intseq(n)] // Marius A. Burtea, Sep 12 2019

a:=proc(n) local nn,j,b,bb: nn:=convert(n,base,10): for j from 1 to nops(nn) do b[j]:=n/nn[j] od: bb:=[seq(b[j],j=1..nops(nn))]: if map(floor,bb)=bb then n else fi end: 1,2,3,4,5,6,7,8,9,seq(seq(seq(a(100*m+10*n+k),k=1..9),n=1..9),m=0..6); # Emeric Deutsch

divByEvryDigitQ[n_] := Block[{id = Union[IntegerDigits[n]]}, Union[ IntegerQ[ #] & /@ (n/id)] == {True}]; Select[ Range[ 487],  divByEvryDigitQ[#] &] (* Robert G. Wilson v, Jun 21 2005 *)
Select[Range[500],FreeQ[IntegerDigits[#],0]&&AllTrue[#/ IntegerDigits[ #], IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 31 2019 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8)); if(v[1]==0, return(0)); for(i=1, #v, if(n%v[i], return(0))); 1 \\ Charles R Greathouse IV, Apr 17 2012

is_A034838(n)=my(d=Set(digits(n)));d[1]&&!forstep(i=#d,1,-1,n%d[i]&&return) \\ M. F. Hasler, Jan 10 2016

A034838_list = []
for g in range(1,4):
    for n in product('123456789',repeat=g):
        s = ''.join(n)
        m = int(s)
        if not any(m % int(d) for d in s):
            A034838_list.append(m) # Chai Wah Wu, Sep 18 2014

for n in range(10**3):
    s = str(n)
    if '0' not in s:
        c = 0
        for i in s:
            if n%int(i):
                c += 1
                break
        if not c:
            print(n,end=', ') # Derek Orr, Sep 19 2014

# finite automaton accepting sequence (see comments in A346267)
from math import gcd
def lcm(a, b): return a * b // gcd(a, b)
def inF(q): return q[0]%q[1] == 0
def delta(q, c): return ((10*q[0]+c)%2520, lcm(q[1], c))
def ok(n):
    q = (0, 1)
    for c in map(int, str(n)):
        if c == 0: return False # computation dies
        else: q = delta(q, c)
    return inF(q)
print(list(filter(ok, range(450)))) # Michael S. Branicky, Jul 18 2021

A067458 Sum of remainders when n is divided by its nonzero digits.

Original entry on oeis.org

0, 0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 1, 0, 3, 0, 1, 2, 7, 4, 3, 0, 1, 2, 0, 3, 2, 0, 3, 8, 3, 0, 1, 2, 4, 0, 1, 6, 8, 0, 5, 0, 1, 2, 5, 6, 0, 3, 3, 5, 9, 0, 1, 2, 3, 4, 5, 0, 5, 6, 9, 0, 1, 2, 4, 6, 5, 10, 0, 7, 9, 0, 1, 2, 5, 4, 5, 8, 10, 0, 9, 0, 1, 2, 3, 6, 5, 6, 13, 10, 0, 0

Offset: 10

Amarnath Murthy, Feb 07 2002

a(n) = 0 for 0 < n 10.

a(A002796(n)) = 0; a(A171492(n)) > 0. - Reinhard Zumkeller, Sep 24 2015

			a(14)= 2 as 1 divides 14 and 2 is the remainder obtained when 14 is divided by 4.

Reinhard Zumkeller, Table of n, a(n) for n = 10..10000

Cf. A002796, A171492.

a067458 n = f 0 n where
   f y 0 = y
   f y x = if d == 0 then f y x' else f (y + mod n d) x'
           where (x', d) = divMod x 10
-- Reinhard Zumkeller, Sep 24 2015

Table[Plus @@ Mod[n, Select[IntegerDigits[n], # != 0 &]], {n, 10, 100}]

Edited and extended by Robert G. Wilson v, Feb 11 2002

A239213 Squares that are divisible by each of their nonzero digits.

Original entry on oeis.org

1, 4, 9, 36, 100, 144, 324, 400, 784, 900, 1024, 1296, 1444, 1764, 2304, 2500, 2916, 3600, 9216, 10000, 10404, 11664, 12100, 12996, 14400, 19044, 22500, 24336, 26244, 28224, 32400, 36864, 39204, 40000, 41616, 44100, 48400, 69696, 78400, 82944, 90000, 93636

Offset: 1

Colin Barker, Mar 12 2014

Intersection of A000290 and A002796.

			39204 is in the sequence because 39204 is divisible by 3, 9, 2 and 4.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A239210, A239211, A239212.

Cf. A000290, A002796.

dnzQ[n_]:=And@@Divisible[n,Select[IntegerDigits[n],#!=0&]]; Select[ Range[ 500]^2,dnzQ] (* Harvey P. Dale, Dec 04 2014 *)

isOK(n) = my(v=vecsort(digits(n^2), , 8)); for(i=1+(v[1]==0), #v, if(n^2%v[i], return(0))); 1
s=[]; for(n=1, 1000, if(isOK(n), s=concat(s, n^2))); s

A239222 Cubes that are divisible by each of their nonzero digits.

Original entry on oeis.org

1, 8, 216, 1000, 2744, 8000, 13824, 74088, 125000, 216000, 512000, 1000000, 1061208, 2000376, 2299968, 2744000, 4741632, 5832000, 8000000, 8242408, 8489664, 9261000, 10941048, 12812904, 13824000, 14886936, 16003008, 19683000, 34012224, 40001688, 42144192

Offset: 1

Colin Barker, Mar 12 2014

Intersection of A000578 and A002796.

			74088 is in the sequence because 74088 is divisible by 4, 7 and 8.

Lars Blomberg, Table of n, a(n) for n = 1..10000

Cf. A239219, A239220, A239221.

Cf. A000578, A002796.

dedQ[n_]:=And@@Divisible[n,Select[IntegerDigits[n],#>0&]]; Select[ Range[ 400]^3, dedQ] (* Harvey P. Dale, Apr 25 2015 *)

isOK(n) = my(v=vecsort(digits(n^3), , 8)); for(i=1+(v[1]==0), #v, if(n^3%v[i], return(0))); 1
s=[]; for(n=1, 1000, if(isOK(n), s=concat(s, n^3))); s

a(n) = A239221^(n)^3. - Michel Marcus, Mar 19 2014

A078546 LCM of n and its nonzero decimal digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 39, 28, 15, 48, 119, 72, 171, 20, 42, 22, 138, 24, 50, 78, 378, 56, 522, 30, 93, 96, 33, 204, 105, 36, 777, 456, 117, 40, 164, 84, 516, 44, 180, 276, 1316, 48, 1764, 50, 255, 260, 795, 540, 55, 840, 1995, 1160, 2655, 60, 366, 186

Offset: 1

Labos Elemer, Dec 05 2002

			n=13: lcm(13,1,3) = 39 = a(13); a(x)=x if x is divisible by each nonzero digits, i.e., if x in A002796.

Cf. A052429, A078547, A078548, A002796.

lc[x_] := Apply[LCM, DeleteCases[IntegerDigits[x], 0]] Table[LCM[lc[w], w], {w, 1, 128}] NOT

lcnzd(n) = lcm(select(x->(x!=0), digits(n)));
a(n) = lcm(n, lcnzd(n)); \\ Michel Marcus, Mar 18 2018

a(n) = lcm(n, A052429(n)).

A171492 Numbers that are not divisible by each of their nonzero decimal digits.

Original entry on oeis.org

13, 14, 16, 17, 18, 19, 21, 23, 25, 26, 27, 28, 29, 31, 32, 34, 35, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 103, 106

Offset: 1

Jaroslav Krizek, Dec 10 2009

Complement of A002796.

A067458(a(n)) > 0. - Reinhard Zumkeller, Sep 24 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Cf. A002796, A034838, A067458.

import Data.List (nub, sort); import Data.Char (digitToInt)
a171492 n = a171492_list !! (n-1)
a171492_list = filter f [1..] where
   f x = any ((> 0) . mod x) ds where
     ds = map digitToInt (if c == '0' then cs else cs')
     cs'@(c:cs) = nub $ sort $ show x
-- Reinhard Zumkeller, Jan 01 2014

sol:=[];for k in [1..120] do a:=Set(Intseq(k)) diff {0}; if #[c:c in a|IsIntegral(k/c)] ne #a then; Append(~sol,k); end if; end for; sol; // Marius A. Burtea, Sep 09 2019

a[c_]:=Module[{b=DeleteCases[IntegerDigits[c], 0]}, !And@@Divisible[c, b]]; Select[Range[250], a] (* Metin Sariyar, Sep 14 2019 *)

a(n) ~ k*n, where k = 2520/2519 = 1.00039.... - Charles R Greathouse IV, Feb 13 2017

A237851 a(1)=1; a(n) is the smallest integer not yet in the sequence divisible by all nonzero digits of a(n-1).

Original entry on oeis.org

1, 2, 4, 8, 16, 6, 12, 10, 3, 9, 18, 24, 20, 14, 28, 32, 30, 15, 5, 25, 40, 36, 42, 44, 48, 56, 60, 54, 80, 64, 72, 70, 7, 21, 22, 26, 66, 78, 112, 34, 84, 88, 96, 90, 27, 98, 144, 52, 50, 35, 45, 100, 11, 13, 33, 39, 63, 102, 38, 120, 46, 108, 104, 68, 168

Offset: 1

Eric Angelini, Feb 14 2014

A permutation of the naturals with inverse A237860.

If a(n) is a prime greater than 7 then no digit of a(n-1) is greater than 1, cf. A007088.

Lars Blomberg and Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A002796, A180410, A002473, A237860, A007088.

import Data.List (nub, sort, delete)
a237851 n = a237851_list !! (n-1)
a237851_list = 1 : f 1 [2..] where
   f x zs = g zs where
     g (u:us) | all ((== 0) . (mod u)) ds = u : f u (delete u zs)
              | otherwise = g us
              where ds = dropWhile (<= 1) $
                         sort $ nub $ map (read . return) $ show x
-- Reinhard Zumkeller, Feb 14 2014

a[1] = 1;
a[n_] := a[n] = For[k = 1, True, k++, If[FreeQ[Array[a, n-1], k], If[ AllTrue[Select[IntegerDigits[a[n-1]], #>0&] // Union, Divisible[k, #]&], Return[k]]]];
a /@ Range[100] (* Jean-François Alcover, Nov 26 2019 *)

A078547 a(n) = lcm(n, A052429(n)) - n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 26, 14, 0, 32, 102, 54, 152, 0, 21, 0, 115, 0, 25, 52, 351, 28, 493, 0, 62, 64, 0, 170, 70, 0, 740, 418, 78, 0, 123, 42, 473, 0, 135, 230, 1269, 0, 1715, 0, 204, 208, 742, 486, 0, 784, 1938, 1102, 2596, 0, 305, 124, 63, 128, 325, 0

Offset: 1

Labos Elemer, Dec 05 2002

a(n)=0 if n is divisible by each nonzero digit, i.e., if n in A002796.

Cf. A052429, A078546, A078548, A002796.

lc[x_] := Apply[LCM, DeleteCases[IntegerDigits[x], 0]] Table[LCM[lc[w], w]-w, {w, 1, 128}]

lcnzd(n) = lcm(select(x->(x!=0), digits(n)));
a(n) = lcm(n, lcnzd(n)) - n; \\ Michel Marcus, Mar 18 2018

cp's OEIS Frontend

A342855 Numbers that contain a digit 0 (A011540) and that are divisible by their nonzero digits (A002796).

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

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A034838 Numbers k that are divisible by every digit of k.

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A067458 Sum of remainders when n is divided by its nonzero digits.

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A239213 Squares that are divisible by each of their nonzero digits.

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A239222 Cubes that are divisible by each of their nonzero digits.

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