A067458 -id:A067458 - cp's OEIS frontend

A002796 Numbers that are divisible by each nonzero digit.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 104, 105, 110, 111, 112, 115, 120, 122, 124, 126, 128, 132, 135, 140, 144, 150, 155, 162, 168, 175, 184, 200, 202, 204, 208, 210, 212

Offset: 1

N. J. A. Sloane

If n is a member so is 10*n. Also all repunits are members. - Robert G. Wilson v, Apr 12 2015
The repdigits are also members because they're always the repunit number of the same length multiplied by the digit being repeated. - Eric Fox, Sep 02 2019
The number of terms < 10^k: 9, 32, 137, 751, 4577, 29950, 207197, 1495637, ... . - Robert G. Wilson v, Apr 13 2015
Includes all multiples of 2520. - Robert Israel, Apr 15 2015
For n >= 10: A067458(a(n)) = 0. - Reinhard Zumkeller, Sep 24 2015

Lindon, Visible factor numbers, J. Rec. Math., 1 (1968), 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..2000
Index entries for 10-automatic sequences.

Cf. A007602, A034838, A237851.
Cf. A171492 (complement).
Cf. A067458.

import Data.List (nub, sort); import Data.Char (digitToInt)
a002796 n = a002796_list !! (n-1)
a002796_list = filter f [1..] where
   f x = all ((== 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..220] do a:=Set(Intseq(k)) diff {0}; if #[c:c in a|IsIntegral(k/c)] eq #a then; Append(~sol,k); end if; end for; sol; // Marius A. Burtea, Sep 09 2019

select(t -> t mod ilcm(op(convert(convert(t,base,10),set) minus {0})) = 0, [$1..1000]); # Robert Israel, Apr 15 2015

dQ[n_]:=Module[{nzidn=DeleteCases[IntegerDigits[n],0]},And@@Divisible[n, nzidn]]; Select[Range[250],dQ] (* Harvey P. Dale, Dec 13 2011 *)

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

A002796_list = []
for i in range(1,10**5):
    for d in set(str(i)):
        if d != '0' and i % int(d):
            break
    else:
        A002796_list.append(i) # Chai Wah Wu, Mar 26 2021

a(n) ~ 2520*n. - Charles R Greathouse IV, Feb 13 2017

More terms from Henry Bottomley, Jun 06 2000

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

A088330 Sum of the remainders when n is divided by nonzero numbers obtained by deleting one digit. The sum ranges over all the 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, 3, 8, 16, 10, 5, 20, 14, 12, 24, 0

Offset: 10

Amarnath Murthy, Oct 01 2003

Differs from A067458 first at n=101, where A067458(101)=0 and a(101) = (101 mod 1) + (101 mod 11) + (101 mod 10) = 0+2+1 = 3. - R. J. Mathar, Sep 11 2008

			a(1234) = Rem[1234/123] + Rem[1234/124]+ Rem[1234/134] + Rem[1234/234] = 4+ 118 + 28 + 64 = 214 where Rem [a/b] = the remainder when a is divided by b.

Robert Israel, Table of n, a(n) for n = 10..10000

Cf. A000042.

f:= proc(n) local L,d,i,j,x,t;
     L:= convert(n,base,10); d:= nops(L); t:= 0;
     for i from 1 to d do
       x:= add(L[j]*10^(j-1),j=1..i-1) + add(L[j]*10^(j-2),j=i+1..d);
       if x <> 0 then t:= t + (n mod x) fi;
     od;
     t
end proc:
map(f, [$10 .. 200]); # Robert Israel, Dec 05 2024

a((10^n - 1)/9) = n. for n > 2. a(1111111 n times ) = a(A000042(n)) = n, n > 2.

a(10 * n) = 10 * a(n). - Robert Israel, Dec 05 2024

More terms from Ray Chandler, Oct 06 2003

cp's OEIS Frontend

A002796 Numbers that are divisible by each nonzero digit.

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

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A088330 Sum of the remainders when n is divided by nonzero numbers obtained by deleting one digit. The sum ranges over all the digits.

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Maple

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