A034837 -id:A034837 - cp's OEIS frontend

A038772 Numbers not divisible by any of their digits.

23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307, 308, 323, 329, 334, 337, 338

Offset: 1

Henry Bottomley, May 04 2000

A038769(a(n)) = 0; complement of A038770.
This is a regular language when written in decimal, though the minimal regular expression is probably thousands of characters long. - Charles R Greathouse IV, Aug 19 2011
Exponential density 0.954... = A104139. Asymptotically 8/35 * n^0.954... + O(n^0.903...) members up to n. - Charles R Greathouse IV, Jul 22 2012

			34 is divisible by neither 3 nor 4.
35 is excluded because 5 is a divisor of 35, but 37 is included because neither 3 nor 7 is a divisor of 37

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences

Cf. A327561 (counts), A038770 (complement).

Cf. also A034709, A034837, A038769.

import Data.Char (digitToInt)
a038772 n = a038772_list !! (n-1)
a038772_list = filter p [1..] where
   p n = all (> 0) $ map ((mod n) . digitToInt) $ filter (> '0') $ show n
-- Reinhard Zumkeller, Jun 19 2011

[k:k in [1..340]| forall{c:c in Set(Intseq(k)) diff {0}|k mod c ne 0}]; // Marius A. Burtea, Dec 22 2019

nddQ[n_]:=Module[{idn=DeleteCases[IntegerDigits[n],0]},And@@Table[ !Divisible[n, idn[[i]]],{i,Length[idn]}]]; Select[Range[350],nddQ] (* Harvey P. Dale, Nov 01 2011 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))), , 8)); for(i=if(v[1], 1, 2), #v, if(n%v[i]==0, return(0))); 1 \\ Charles R Greathouse IV, Jul 22 2011

def ok(n): return not any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 339)))) # Michael S. Branicky, May 20 2021

Edited by N. J. A. Sloane, Nov 17 2008 at the suggestion of R. J. Mathar

A038770 Numbers divisible by at least one of their digits.

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, 25, 26, 28, 30, 31, 32, 33, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 55, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 75, 77, 80, 81, 82, 84, 85, 88, 90, 91, 92, 93, 95, 96, 99, 100, 101, 102

Offset: 1

Henry Bottomley, May 04 2000

A038769(a(n)) > 0; complement of A038772.

The decimal digit strings of this sequence are a regular language, since it is the union of A011531 and A121022 .. A121029 which are likewise regular languages. Some computer state machine manipulation for this union shows a minimum deterministic finite automaton (DFA) matching the digit strings of this sequence has 11561 states. Reversing strings (so least significant digit first) reduces to 1699 states, or reverse and allow high 0's (which become trailing 0's due to the reverse) reduces to 1424 states. The latter are tractable sizes for the linear recurrence in A327560. - Kevin Ryde, Dec 04 2019

			35 is included because 5 is a divisor of 35, but 37 is not included because neither 3 nor 7 is a divisor of 37.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences

Cf. A327560 (counts), A038772 (complement), A034709, A034837, A038769.

a038770 n = a038770_list !! (n-1)
a038770_list = filter f [1..] where
   f u = g u where
     g v = v > 0 && (((d == 0 || r > 0) && g v') || r == 0)
           where (v',d) = divMod v 10; r = mod u d
-- Reinhard Zumkeller, Jul 30 2015, Jun 19 2011

Select[Range[120],MemberQ[Divisible[#,Cases[IntegerDigits[#],Except[0]]], True]&] (* Harvey P. Dale, Jun 20 2011 *)
Select[Range[120],AnyTrue[#/DeleteCases[IntegerDigits[#],0],IntegerQ]&] (* Harvey P. Dale, Mar 29 2024 *)

is(n)=my(v=vecsort(eval(Vec(Str(n))),,8));for(i=if(v[1],1,2),#v,if(n%v[i]==0,return(1)));0 \\ Charles R Greathouse IV, Jul 22 2011

def ok(n): return any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 103)))) # Michael S. Branicky, May 20 2021

a(n) ~ n. - Charles R Greathouse IV, Jul 22 2011

A038769 Number of digits of n which are divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 0, 0, 1, 1, 1, 1, 0, 2, 1, 0, 0, 2, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 1, 1, 1, 0, 0, 1, 0, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 2, 1, 2, 2, 1, 2, 2

Offset: 1

Henry Bottomley, May 04 2000

a(A038772(n)) = 0; a(A038770(n)) > 0.

			a(35)=1 because 5 is a divisor of 35 but 3 is not.

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

Cf. A034709, A034837, A038770, A038772.

import Data.Char (digitToInt)
a038769 n = length $ filter (== 0)
            $ map ((mod n) . digitToInt) $ filter (> '0') $ show n
-- Reinhard Zumkeller, Jun 19 2011

[#[c:c in Intseq(k) |not IsZero(c) and k mod c eq 0]:k in [1..105]]; // Marius A. Burtea, Dec 23 2019

f:= proc(n) local L; L:= convert(n,base,10);
  nops(select(t -> t > 0 and n mod t = 0, L))
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 04 2016

Array[Count[Position[Most@ DigitCount@ #, ?(# > 0 &)][[All, 1]], k /; Mod[#, k] == 0] &, 105] (* Michael De Vlieger, Dec 23 2019 *)
Table[Count[n/Select[IntegerDigits[n],#>0&],?IntegerQ],{n,110}] (* _Harvey P. Dale, Mar 04 2023 *)

A132359 Numbers divisible by the square of their last decimal digit.

Original entry on oeis.org

1, 11, 12, 21, 25, 31, 32, 36, 41, 51, 52, 61, 63, 64, 71, 72, 75, 81, 91, 92, 101, 111, 112, 121, 125, 128, 131, 132, 141, 144, 147, 151, 152, 153, 161, 171, 172, 175, 181, 191, 192, 201, 211, 212, 216, 221, 224, 225, 231, 232, 241, 243, 251, 252, 261, 271, 272

Offset: 1

Jonathan Vos Post, Nov 08 2007

Subsequences are A017281 and A053742 representing last digits 1 and 5. Generators for the subsequences representing last digits 2, 3, 4, 6, 7, 8 and 9 are, in that order, the terms 12+20i, 63+90i, 64+80i, 36+180i, 147+490i, 128+320i, 729+810i, where i=0,1,2,... - R. J. Mathar, Nov 13 2007

This is a 10-automatic sequence. - Charles R Greathouse IV, Dec 28 2011

			147 belongs to the sequence because 147/7^2 = 3.

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 values from T. D. Noe)
Index entries for 10-automatic sequences.

Cf. A034709, A225722, A221651, A225296, A225297, A034709, A034837, A005349, A007602, A034838, A225297.

isA132359 := proc(n) local ldig ; ldig := n mod 10 ; if ldig <> 0 and n mod (ldig^2) = 0 then true ; else false ; fi ; end: for n from 1 to 400 do if isA132359(n) then printf("%d,",n) ; fi ; od: # R. J. Mathar, Nov 13 2007
a:=proc(n) local nn: nn:=convert(n,base,10): if 0 < nn[1] and `mod`(n,nn[1]^2) =0 then n else end if end proc: seq(a(n),n=1..250); # Emeric Deutsch, Nov 15 2007

Select[Range[250], IntegerDigits[ # ][[ -1]] > 0 && Mod[ #, IntegerDigits[ # ][[ -1]]^2] == 0 &] (* Stefan Steinerberger, Nov 12 2007 *)
dsldQ[n_]:=Module[{lidnsq=Last[IntegerDigits[n]]^2},lidnsq!=0 && Divisible[n,lidnsq]]; Select[Range[300],dsldQ] (* Harvey P. Dale, May 03 2011 *)

is(n)=n%(n%10)^2==0 \\ Charles R Greathouse IV, Dec 28 2011

def ok(n): return n%10 > 0 and n%(n%10)**2 == 0
print([k for k in range(273) if ok(k)]) # Michael S. Branicky, Jul 03 2022

which(sapply(1:500,function(x) isint(x/(x%%10)^2))) # Christian N. K. Anderson, May 04 2013

Numbers k such that fp[k / (k mod 10)] = 0.

a(n) ~ 6350400*n/1241929 = 5.113...*n. - Charles R Greathouse IV, Dec 28 2011

Corrected and extended by Stefan Steinerberger, Emeric Deutsch and R. J. Mathar, Nov 12 2007

A178157 Numbers k that are divisible by every prefix of k.

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, 22, 24, 26, 28, 30, 33, 36, 39, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 330, 360, 390, 400, 420, 440, 460, 480, 500, 510, 520, 530, 540, 550, 560

Offset: 1

Michel Lagneau, Dec 17 2010

			3570 is in the sequence because:
     3 | 3570;
    35 | 3570;
   357 | 3570;
  3570 | 3570.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A034837.

with(numtheory):T:=array(1..9):for n from 1 to 10000 do:ind:=0:l:=length(n):n0:=n:s:=0:for
  m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :T[m]:=u:od:for i from
  1 to l do: s1:=0:for j from 0 to i-1 do: s1:=s1 + T[l-i+j+1]*10^j :od:if irem(n,s1)=0
  then ind:=ind+1:else fi:od:if ind=l then printf(`%d, `,n):else fi:od:

Select[Range[560],AllTrue[Table[Boole[Divisible[#,Floor[#/10^i]]],{i,0,Log10[#]}],Positive] &] (* Stefano Spezia, May 03 2025 *)

A221651 Numbers divisible by their first digit squared (excluding those whose first digit is 1).

Original entry on oeis.org

20, 24, 28, 36, 48, 50, 200, 204, 208, 212, 216, 220, 224, 228, 232, 236, 240, 244, 248, 252, 256, 260, 264, 268, 272, 276, 280, 284, 288, 292, 296, 306, 315, 324, 333, 342, 351, 360, 369, 378, 387, 396, 400, 416, 432, 448, 464, 480, 496, 500, 525, 550, 575

Offset: 1

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

Numbers where floor(n/10^floor(log(n)))^2 divides n.

			48 is divisible by 4^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000
Christian N. K. Anderson, Ulam Spiral for the first 10000 terms. [Draw the usual Ulam spiral, containing all the nonnegative integers, and color the numbers belonging to this sequence yellow]

Cf. A132359, A034709, A034837, A005349, A007602, A034838, A225297, A225722, A225683.

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<10000) if((fd=firstdig((x=x+1)))>1) if(isint(x/fd^2)) y[(len=len+1)]=x

A225297 Numbers divisible by their last digit cubed.

Original entry on oeis.org

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

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) ))

A225299 Numbers divisible by the square of each digit.

Original entry on oeis.org

1, 11, 12, 36, 111, 112, 128, 144, 212, 216, 224, 333, 432, 448, 612, 1111, 1112, 1116, 1212, 1296, 1332, 1424, 2112, 2144, 2212, 2224, 2232, 2916, 3132, 3312, 3636, 4112, 4144, 4224, 4288, 4464, 6336, 6624, 8128, 8448, 9396, 11111, 11112, 11133, 11172, 11212

Offset: 1

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

Includes all repunits.

			a(7) 128 is divisible by 1^2, by 2^2, and by 8^2.

Christian N. K. Anderson, Table of n, a(n) for n = 1..5000

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

d[n_]:=IntegerDigits[n]; t={}; Do[If[!MemberQ[d[n],0] && Union[Mod[n,d[n]^2]] == {0}, AppendTo[t,n]], {n,11220}]; t (* Jayanta Basu, May 15 2013 *)
Select[Range[12000],DigitCount[#,10,0]==0&&And@@Divisible[ #,IntegerDigits[ #]^2]&] (* Harvey P. Dale, Jul 16 2018 *)

isint<-function(x) x==as.integer(x)
sqalldig<-function(x) as.numeric(strsplit(as.character(x),"")[[1]])^2
divby<-function(x) ifelse(length(grep(0,x))>0,F,all(isint(x/sqalldig(x))))
which(sapply(1:1000,divby))

cp's OEIS Frontend

A038772 Numbers not divisible by any of their digits.

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A038770 Numbers divisible by at least one of their digits.

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A038769 Number of digits of n which are divisors of n.

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A132359 Numbers divisible by the square of their last decimal digit.

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A178157 Numbers k that are divisible by every prefix of k.

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A221651 Numbers divisible by their first digit squared (excluding those whose first digit is 1).

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A225297 Numbers divisible by their last digit cubed.