cp's OEIS Frontend

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

%I A034838 #111 Feb 16 2025 08:32:37
%S A034838 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 A034838 115,122,124,126,128,132,135,144,155,162,168,175,184,212,216,222,224,
%U A034838 244,248,264,288,312,315,324,333,336,366,384,396,412,424,432,444,448
%N A034838 Numbers k that are divisible by every digit of k.
%C A034838 Subset of zeroless numbers A052382: Integers with at least one digit 0 (A011540) are excluded.
%C A034838 A128635(a(n)) = n.
%C A034838 Contains in particular all repdigits A010785 \ {0}. - _M. F. Hasler_, Jan 05 2020
%C A034838 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 A034838 Named "nude numbers" by Katagiri (1982-83). - _Amiram Eldar_, Jun 26 2021
%D A034838 Charles Ashbacher, Journal of Recreational Mathematics, Vol. 33 (2005), pp. 227. See problem number 2693.
%D A034838 Yoshinao Katagiri, Letter to the editor of the Journal of Recreational Mathematics, Vol. 15, No. 4 (1982-83).
%D A034838 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 A034838 Thomas Koshy, Elementary Number Theory with Applications, Elsevier, 2007, p. 79.
%H A034838 Reinhard Zumkeller, <a href="/A034838/b034838.txt">Table of n, a(n) for n = 1..10000</a>
%H A034838 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 A034838 Giovanni Resta, <a href="https://www.numbersaplenty.com/set/nude_number/">nude numbers</a>, Numbersaplenty, 2013.
%H A034838 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 A034838 Voodooguru, <a href="https://voodooguru23.blogspot.com/2020/10/nude-numbers.html">Nude Numbers</a>, Mathematical Meanderings, Oct 11 2020.
%H A034838 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Digit.html">Digit</a>
%H A034838 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%e A034838 36 is in the sequence because it is divisible by both 3 and 6.
%e A034838 48 is included because both 4 and 8 divide 48.
%e A034838 64 is not included because even though 4 divides 64, 6 does not.
%p A034838 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 A034838 divByEvryDigitQ[n_] := Block[{id = Union[IntegerDigits[n]]}, Union[ IntegerQ[ #] & /@ (n/id)] == {True}]; Select[ Range[ 487],  divByEvryDigitQ[#] &] (* _Robert G. Wilson v_, Jun 21 2005 *)
%t A034838 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 A034838 (Haskell)
%o A034838 a034838 n = a034838_list !! (n-1)
%o A034838 a034838_list = filter f a052382_list where
%o A034838    f u = g u where
%o A034838      g v = v == 0 || mod u d == 0 && g v' where (v',d) = divMod v 10
%o A034838 -- _Reinhard Zumkeller_, Jun 15 2012, Dec 21 2011
%o A034838 (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 A034838 (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 A034838 (Python)
%o A034838 A034838_list = []
%o A034838 for g in range(1,4):
%o A034838     for n in product('123456789',repeat=g):
%o A034838         s = ''.join(n)
%o A034838         m = int(s)
%o A034838         if not any(m % int(d) for d in s):
%o A034838             A034838_list.append(m) # _Chai Wah Wu_, Sep 18 2014
%o A034838 (Python)
%o A034838 for n in range(10**3):
%o A034838     s = str(n)
%o A034838     if '0' not in s:
%o A034838         c = 0
%o A034838         for i in s:
%o A034838             if n%int(i):
%o A034838                 c += 1
%o A034838                 break
%o A034838         if not c:
%o A034838             print(n,end=', ') # _Derek Orr_, Sep 19 2014
%o A034838 (Python) # finite automaton accepting sequence (see comments in A346267)
%o A034838 from math import gcd
%o A034838 def lcm(a, b): return a * b // gcd(a, b)
%o A034838 def inF(q): return q[0]%q[1] == 0
%o A034838 def delta(q, c): return ((10*q[0]+c)%2520, lcm(q[1], c))
%o A034838 def ok(n):
%o A034838     q = (0, 1)
%o A034838     for c in map(int, str(n)):
%o A034838         if c == 0: return False # computation dies
%o A034838         else: q = delta(q, c)
%o A034838     return inF(q)
%o A034838 print(list(filter(ok, range(450)))) # _Michael S. Branicky_, Jul 18 2021
%o A034838 (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 A034838 Intersection of A002796 (numbers divisible by each nonzero digit) and A052382 (zeroless numbers), or A002796 \ A011540 (numbers with digit 0).
%Y A034838 Subsequence of A034709 (divisible by last digit).
%Y A034838 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 A034838 Cf. A113028, A346267 (number of terms with n digits), A087140 (complement).
%Y A034838 Supersequence of A115569 (with all different digits).
%K A034838 nonn,base,nice
%O A034838 1,2
%A A034838 _Erich Friedman_