A034838 -id:A034838 - cp's OEIS frontend

A225297 Numbers divisible by their last digit cubed.

1, 11, 21, 31, 32, 41, 51, 61, 64, 71, 72, 81, 91, 101, 111, 112, 121, 125, 131, 141, 151, 152, 161, 171, 181, 191, 192, 201, 211, 216, 221, 231, 232, 241, 243, 251, 261, 271, 272, 281, 291, 301, 311, 312, 321, 331, 341, 351, 352, 361, 371, 375, 381, 384, 391

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers k such that (k mod 10)^3 | k.
All numbers ending in 1 are trivially included in this sequence.
The sequence is { 1+10k, 32 + 40k, 243 + 270k, 64 + 320k, 125 + 250k, 216 + 1080k, 3087 + 3430k, 2048 + 2560k, 729 + 7290k ; k = 0,1,2,...}. - M. F. Hasler, Jan 31 2016
The asymptotic density of this sequence is 2201597407/16003008000 = 0.137573... . - Amiram Eldar, Aug 08 2023

			a(16) = 112 is divisible by 2^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

Select[Range[400], (m = Mod[#, 10]) > 0 && Divisible[#, m^3] &] (* Amiram Eldar, Aug 08 2023 *)

is(n)=n%10&&n%(n%10)^3==0 \\ M. F. Hasler, Jan 31 2016

x=0; y=rep(0,100); len=0; isint<-function(x) x==as.integer(x); while(len<100) if((x=x+1)%%10>0) if(isint(x/(x%%10)^3)) y[(len=len+1)]=x

A263314 Numbers m whose decimal representation includes at least one digit that divides every digit of m.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 26, 28, 30, 31, 33, 36, 39, 40, 41, 42, 44, 48, 50, 51, 55, 60, 61, 62, 63, 66, 70, 71, 77, 80, 81, 82, 84, 88, 90, 91, 93, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112

Offset: 1

Giovanni Teofilatto, Oct 14 2015

Every number that has a digit '1' (cf. A011531) is in this sequence. Sequence A034838 (every digit divides n) is not a subsequence (e.g., 324 is not in the present sequence). - M. F. Hasler, Jan 10 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

is(n)={n && (n=select(t->t,Set(digits(n))))%n[1]==0} \\ The divisor is necessarily the smallest nonzero digit. A vector having no nonzero component is considered equal to 0. - M. F. Hasler, Jan 10 2016

A263314_list = []
for i in range(10**4):
    s = str(i)
    for d in s:
        j = int(d)
        if j :
            for e in s:
                if int(e) % j:
                    break
            else:
                A263314_list.append(i)
                break
# Chai Wah Wu, Oct 21 2015

A346576 Let x run through the list of numbers with no zeros (A052382); replace each digit d of x by the digit (x mod d).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 4, 3, 2, 1, 10, 0, 12, 0, 10, 2, 16, 4, 12, 10, 20, 0, 12, 20, 0, 12, 26, 3, 10, 20, 31, 0, 10, 24, 35, 0, 14, 10, 20, 32, 42, 0, 12, 21, 32, 45, 10, 20, 30, 40, 50, 0, 14, 24, 36, 10, 20, 31, 42, 50, 64, 0, 16, 27, 10

Offset: 1

Rakesh Khanna A, Jul 24 2021

Graph of the sequence generates a fractal-like image.

			If x = 247 we get 132 as 247 mod 2 = 1, 247 mod 4 = 3, and 247 mod 7 = 2. As 247 is the 205th zeroless number,  a(205) =  132.

Michel Marcus, Table of n, a(n) for n = 1..7380
Rakesh Khanna A, Graph of the sequence

Cf. A034838, A052382.

See A347323 for another version.

#include 
#define START 1
#define END 1000
int main(){
  unsigned int R,N,M,power_cntr;
  int mod1,mod2;
  for(N=START;N<=END;N++){
    R=N;
    M=0;
    power_cntr=1;
    while(R!=0){
      mod1=R%10;
      if(mod1==0) break;
      mod2=N%mod1;
      M+=mod2*power_cntr;
      power_cntr*=10;
      R=R/10;}
    if(mod1!=0) printf("%u %u\n",N,M);}
  return 0;}

f[n_] := FromDigits @ Mod[n, IntegerDigits[n]]; f /@ Select[Range[100], !MemberQ[IntegerDigits[#], 0] &] (* Amiram Eldar, Jul 26 2021 *)

a(m) = my(d=digits(m)); fromdigits(Vec(apply(x->(m % x), d)));
apply(x->a(x), select(x->vecmin(digits(x)), [1..100])) \\ Michel Marcus, Jul 24 2021

def f(k, digits): return int("".join(map(str, map(lambda x: k%x, digits))))
def aupton(terms):
    alst, k = [], 1
    while len(alst) < terms:
        s = str(k)
        if '0' not in s: alst.append(f(k, list(map(int, s))))
        k += 1
    return alst
print(aupton(73)) # Michael S. Branicky, Aug 22 2021

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

A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

Original entry on oeis.org

24, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 324, 351, 378, 448, 500, 648, 2000, 2008, 2016, 2024, 2032, 2040, 2048, 2056, 2064, 2072, 2080, 2088, 2096, 2104, 2112, 2120, 2128, 2136, 2144, 2152, 2160, 2168, 2176, 2184, 2192, 2200, 2208

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 04 2013

Numbers where floor(n/10^floor(log(n)))^3 | n.

			448 is divisible by 4^3.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral of first 10000 terms.

Cf. A132359, A034709, A034837, A005349, A007602, A034838.

dfdcQ[n_]:=Module[{fidn=IntegerDigits[n][[1]]},fidn!=1&&Divisible[ n,fidn^3]]; Select[Range[2500],dfdcQ] (* Harvey P. Dale, Jan 13 2019 *)

x=0; y=rep(0,1000); len=0
firstdig<-function(x) as.numeric(substr(as.character(x),1,1))
isint<-function(x) x==as.integer(x)
while(len<1000) if((fd=firstdig((x=x+1)))>1) if(isint(x/fd^3)) y[(len=len+1)]=x

A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

Original entry on oeis.org

32, 64, 72, 112, 125, 152, 192, 216, 232, 243, 272, 312, 352, 375, 384, 392, 432, 472, 512, 513, 552, 592, 625, 632, 672, 704, 712, 729, 752, 783, 792, 832, 872, 875, 912, 952, 992, 1024, 1032, 1053, 1072, 1112, 1125, 1152, 1192, 1232, 1272, 1296, 1312, 1323

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 13 2013

a(n) ~ n. For 69 < n < 10000, the formula 26.61*n - 2.76 provides an estimate of a(n) to within 1%.

The asymptotic density of this sequence is 601296607/16003008000 = 0.037573... . Therefore, contrary to the above comment, a(n) ~ c*n where c = 16003008000/601296607 = 26.614166... . - Amiram Eldar, Aug 08 2023

			a(5) = 125 is an example because its last digit is 5, and 5^3 = 125, and 125 is divisible by 125.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Subsequence of A225297.

Cf. A221651, A225296, A132359, A034709, A034837, A005349, A007602, A034838.

dldcQ[n_]:=Module[{ld=Last[IntegerDigits[n]]},ld>1&&Divisible[n,ld^3]]; Select[Range[1500],dldcQ] (* Harvey P. Dale, Aug 15 2014 *)

which(sapply(1:1000,function(x) x%%10>1 & (v=x/(x%%10)^3)==as.integer(v) ))

A337163 Numbers divisible by their individual digits, but not by the product of their digits.

Original entry on oeis.org

22, 33, 44, 48, 55, 66, 77, 88, 99, 122, 124, 126, 155, 162, 168, 184, 222, 244, 248, 264, 288, 324, 333, 336, 366, 396, 412, 424, 444, 448, 488, 515, 555, 636, 648, 666, 728, 777, 784, 824, 848, 864, 888, 936, 999, 1122, 1124, 1128, 1144, 1155, 1164, 1222

Offset: 1

Bernard Schott, Jan 28 2021

The sequence is infinite. For example, all numbers of the form ((10^n-1)/9)*(10^2)+24 are terms for n > 0. The numbers of this form will never be divisible by 8 but they will always be divisible by 1, 2 and 4. Also there are infinitely many terms any three of whose consecutive digits are distinct, for example, concatenations of 124. Are there infinitely many terms which don't consist of periodically repeating substrings? - Metin Sariyar, Jan 28 2021

Every repdigit non-repunit with at least 2 digits is a term. - Bernard Schott, Jan 28 2021

			48 is divisible by 4 and 8, but 48 is not divisible by 4*8 = 32, so 48 is a term.
128 is divisible by 1, 2 and 8, and 128 is divisible by 1*2*8 = 16 with 128 = 16*8, so 128 is not a term.

David A. Corneth, Table of n, a(n) for n = 1..10000

Intersection of A034838 and A188643.
Cf. A087142 (similar, with sum).
Cf. A087141, A285271.

q[n_] := AllTrue[(digits = IntegerDigits[n]), # > 0 && Divisible[n, #] &] && !Divisible[n, Times @@ digits]; Select[Range[1000], q] (* Amiram Eldar, Jan 28 2021 *)

isok(n) = my(d=digits(n)); if (vecmin(d), for (i=1, #d, if (n % d[i], return(0))); (n % vecprod(d))); \\ Michel Marcus, Jan 28 2021

More terms from Michel Marcus, Jan 28 2021

A087140 Numbers that are not divisible by all their digits in decimal representation.

Original entry on oeis.org

10, 13, 14, 16, 17, 18, 19, 20, 21, 23, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91

Offset: 1

Reinhard Zumkeller, Aug 18 2003

Complement of A034838.

Cf. A087141.

A128635 Number of numbers not greater than n and divisible by all digits in decimal representation.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20

Offset: 1

Reinhard Zumkeller, Mar 16 2007

a(A034838(n)) = n.

			a(42) = #{1,2,3,4,5,6,7,8,9,11,12,15,22,24,33,36} = 16.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Accumulate[Table[If[DigitCount[n,10,0]==0&&Union[Mod[n,IntegerDigits[n]]] == {0},1,0],{n,80}]] (* Harvey P. Dale, Feb 21 2020 *)

A209933 Numbers that are divisible by all digits of their divisors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 44, 48, 55, 66, 77, 88, 99, 132, 264

Offset: 1

Jaroslav Krizek, Apr 16 2012

There are 24 terms < 10000. Conjecture: next term a(25) = prime repunit with 19 digits 1.
Supersequence of A004022 (prime repunits). Subsequence of A034838 (numbers k that are divisible by every digit of k).
Any further terms are > 10^11. - Lucas A. Brown, Sep 29 2024

			Number 264 with divisors 1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264 is in sequence because all possible digits its divisors (1, 2, 3, 4, 6, 8) are its divisors.

Lucas A. Brown, Python program.

Cf. A004022, A034838, A037278.

Cf. A027750.

import Data.List (unfoldr, nub, sort)
a209933 n = a209933_list !! (n-1)
a209933_list = filter f [1..] where
   f x = head (ds x) /= 0 && all (== 0) (map ((mod x)) (ds x)) where
     ds = sort . nub . concatMap (unfoldr (\z ->
          if z == 0 then Nothing else Just $ swap $ divMod z 10)) .
          a027750_row
-- Reinhard Zumkeller, Apr 19 2012

cp's OEIS Frontend

A225297 Numbers divisible by their last digit cubed.

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A263314 Numbers m whose decimal representation includes at least one digit that divides every digit of m.

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A346576 Let x run through the list of numbers with no zeros (A052382); replace each digit d of x by the digit (x mod d).

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

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A225296 Numbers divisible by their first digit cubed (excluding those whose first digit is 1).

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A225722 Numbers divisible by their last digit cubed, excluding those whose last digit is 1.

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A337163 Numbers divisible by their individual digits, but not by the product of their digits.

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