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

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

1,2

Comments

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

References

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.

Links

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.

Example

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.

Maple

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

Mathematica

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 *)

Prog

(Haskell)
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
(PARI) 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
(PARI) 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
(Python)
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
(Python)
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
(Python) # 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
(Magma) [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

Crossrefs

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).

Sequence in context: A034709 A178158 A337184 * A063527 A209933 A182183

Adjacent sequences: A034835 A034836 A034837 * A034839 A034840 A034841

Keyword

nonn,base,nice

Author

Erich Friedman

Status

approved