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Numbers k that are divisible by every digit of k.
38

%I #110 Jan 20 2024 04:09:51

%S 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,

%T 115,122,124,126,128,132,135,144,155,162,168,175,184,212,216,222,224,

%U 244,248,264,288,312,315,324,333,336,366,384,396,412,424,432,444,448

%N Numbers k that are divisible by every digit of k.

%C Subset of zeroless numbers A052382: Integers with at least one digit 0 (A011540) are excluded.

%C A128635(a(n)) = n.

%C Contains in particular all repdigits A010785 \ {0}. - _M. F. Hasler_, Jan 05 2020

%C 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

%C Named "nude numbers" by Katagiri (1982-83). - _Amiram Eldar_, Jun 26 2021

%D Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227. See problem number 2693.

%D Yoshinao Katagiri, Letter to the editor of the Journal of Recreational Mathematics, Vol. 15, No. 4 (1982-83).

%D 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.

%D Thomas Koshy, Elementary Number Theory with Applications, Elsevier, 2007, p. 79.

%H Reinhard Zumkeller, <a href="/A034838/b034838.txt">Table of n, a(n) for n = 1..10000</a>

%H Diophante, <a href="http://www.diophante.fr/problemes-par-themes/arithmetique-et-algebre/a1-pot-pourri/923-a1916-le-plus-grand-entier-divisible-par-ses-propres-chiffres">A1916. Le plus grand entier divisible par ses propres chiffres</a> (in French).

%H Giovanni Resta, <a href="https://www.numbersaplenty.com/set/nude_number/">nude numbers</a>, Numbersaplenty, 2013.

%H Roberto A. Ribas, <a href="https://www.kappamuepsilon.org/Pentagon/Vol_45_Num_1_Fall_1985.pdf">The Nude Numbers</a>, The Pentagon, Vol. 45, No. 1 (1985), pp. 18-31.

%H Voodooguru, <a href="https://voodooguru23.blogspot.com/2020/10/nude-numbers.html">Nude Numbers</a>, Mathematical Meanderings, Oct 11 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Digit.html">Digit</a>

%H <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.

%e 36 is in the sequence because it is divisible by both 3 and 6.

%e 48 is included because both 4 and 8 divide 48.

%e 64 is not included because even though 4 divides 64, 6 does not.

%p 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_

%t divByEvryDigitQ[n_] := Block[{id = Union[IntegerDigits[n]]}, Union[ IntegerQ[ #] & /@ (n/id)] == {True}]; Select[ Range[ 487], divByEvryDigitQ[#] &] (* _Robert G. Wilson v_, Jun 21 2005 *)

%t 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 *)

%o (Haskell)

%o a034838 n = a034838_list !! (n-1)

%o a034838_list = filter f a052382_list where

%o f u = g u where

%o g v = v == 0 || mod u d == 0 && g v' where (v',d) = divMod v 10

%o -- _Reinhard Zumkeller_, Jun 15 2012, Dec 21 2011

%o (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

%o (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

%o (Python)

%o A034838_list = []

%o for g in range(1,4):

%o for n in product('123456789',repeat=g):

%o s = ''.join(n)

%o m = int(s)

%o if not any(m % int(d) for d in s):

%o A034838_list.append(m) # _Chai Wah Wu_, Sep 18 2014

%o (Python)

%o for n in range(10**3):

%o s = str(n)

%o if '0' not in s:

%o c = 0

%o for i in s:

%o if n%int(i):

%o c += 1

%o break

%o if not c:

%o print(n,end=', ') # _Derek Orr_, Sep 19 2014

%o (Python) # finite automaton accepting sequence (see comments in A346267)

%o from math import gcd

%o def lcm(a, b): return a * b // gcd(a, b)

%o def inF(q): return q[0]%q[1] == 0

%o def delta(q, c): return ((10*q[0]+c)%2520, lcm(q[1], c))

%o def ok(n):

%o q = (0, 1)

%o for c in map(int, str(n)):

%o if c == 0: return False # computation dies

%o else: q = delta(q, c)

%o return inF(q)

%o print(list(filter(ok, range(450)))) # _Michael S. Branicky_, Jul 18 2021

%o (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

%Y Intersection of A002796 (numbers divisible by each nonzero digit) and A052382 (zeroless numbers), or A002796 \ A011540 (numbers with digit 0).

%Y Subsequence of A034709 (divisible by last digit).

%Y 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).

%Y Cf. A113028, A346267 (number of terms with n digits), A087140 (complement).

%Y Supersequence of A115569 (with all different digits).

%K nonn,base,nice

%O 1,2

%A _Erich Friedman_