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
Examples
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.
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.
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).
Supersequence of A115569 (with all different digits).
Programs
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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 -
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
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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
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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 *) -
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
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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
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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
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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
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