A034709 -id:A034709 - cp's OEIS frontend

A038770 Numbers divisible by at least one of their digits.

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

A115569 Lynch-Bell numbers: numbers n such that the digits are all different (and do not include 0) and n is divisible by each of its individual digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 15, 24, 36, 48, 124, 126, 128, 132, 135, 162, 168, 175, 184, 216, 248, 264, 312, 315, 324, 384, 396, 412, 432, 612, 624, 648, 672, 728, 735, 784, 816, 824, 864, 936, 1236, 1248, 1296, 1326, 1362, 1368, 1395, 1632, 1692, 1764, 1824

Offset: 1

Mike Smith (mtm_king(AT)yahoo.com), Mar 10 2006; also submitted by Andy Edwards (AndynGen(AT)aol.com), Mar 20 2006

This is a subset of some of the related sequences listed below. Stephen Lynch and Andrew Bell are Brisbane surgeons who contributed to the identification of this sequence.
There are 548 Lynch-Bell numbers. A117911 gives the number of n-digit ones. The digit 5 cannot appear in Lynch-Bell numbers containing an even digit; 5 must be the units digit when it appears. The 7-digit Lynch-Bell numbers are 105 permutations of 1289736 (the smallest such). - Rick L. Shepherd, Apr 01 2006
Can be seen/read as a table with row lengths A117911 (rows r > 7 have zero length). - M. F. Hasler, Jan 31 2016

			384/3 = 128, 384/8 = 48, 384/4 = 96. Thus 384 is Lynch-Bell as it is a multiple of each of its three distinct digits.

Rick L. Shepherd, List of all terms

Cf. A034838, A034709, A063527.

Cf. A117911, A117912 (have even digits only), A117913 (have odd digits only), A010784.

with(combinat):
f:= l-> parse(cat(l[])):
T:= n-> sort(map(f, select(l-> andmap(x-> irem(f(l), x)=0, l),
         map(p-> permute(p)[], choose([$1..9], n)))))[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Jul 31 2022

Reap[For[n = 1, n < 10^7, n++, id = IntegerDigits[n]; If[FreeQ[id, 0] && Length[id] == Length[Union[id]] && And @@ (Divisible[n, #]& /@ id), Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Nov 26 2013 *)
bnQ[n_]:=Max[DigitCount[n]]==1&&FreeQ[IntegerDigits[n],0]&&Union[Divisible[n,IntegerDigits[ n]]]=={True}; Select[Range[2000],lbnQ] (* Harvey P. Dale, Jun 02 2023 *)
Cases[Union @@ ((FromDigits@#&/@Flatten[Permutations@# & /@ Subsets[Range@9, {#}], 1])&/@ Range@9), ?(DeleteDuplicates[Divisible[#, IntegerDigits@#]] == {True} &)] (* _Hans Rudolf Widmer, Aug 27 2024 *)

A115569_row(n)={if(n,my(u=vectorv(n,i,10^i)\10,S=List(),M);forvec(v=vector(n,i,[1,9]),(M=lcm(v))%10==0||normlp(v,1)%3^valuation(M,3)||for(k=1,n!,vecextract(v,numtoperm(n,k))*u%M ||listput(S,vecextract(v,numtoperm(n,k))*u)),2);Set(S),concat(apply(A115569_row,[1..7])))} \\ Return terms of length n if given, else the vector of all terms. The checks M%10 and |v| % 3^v(...) are not needed but reduce CPU time by 97%. - M. F. Hasler, Jan 31 2016

A115569(n)=n>9&&for(r=2,7,(n-=#t=A115569_row(r))>9||return(t[n-9+#t]));n \\ M. F. Hasler, Jan 31 2016

def ok(n):
    s = str(n)
    if "0" in s or len(set(s)) < len(s): return False
    return all(n%int(d) == 0 for d in s)
afull = [k for k in range(9867313) if ok(k)]
print(afull[:55]) # Michael S. Branicky, Jul 31 2022

The full list of terms was sent in by Rick L. Shepherd (see link) and also by Sébastien Dumortier, Apr 04 2006

A139222 a(n) = 30*n - 27.

Original entry on oeis.org

3, 33, 63, 93, 123, 153, 183, 213, 243, 273, 303, 333, 363, 393, 423, 453, 483, 513, 543, 573, 603, 633, 663, 693, 723, 753, 783, 813, 843, 873, 903, 933, 963, 993, 1023, 1053, 1083, 1113, 1143, 1173, 1203, 1233, 1263, 1293, 1323, 1353, 1383, 1413, 1443, 1473

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 3 with the units digit equal to 3.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139245, A017329, A139249, A139264, A139279 and A139280.

[30*n - 27: n in [1..60]]; // Vincenzo Librandi, Jun 17 2011

Range[3, 2000, 30] (* Vladimir Joseph Stephan Orlovsky, Jun 15 2011 *)
30*Range[50]-27 (* or *) NestList[30+#&,3,50] (* Harvey P. Dale, Mar 27 2021 *)

a(n)=30*n-27 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 30.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 3*x*(1+9*x)/(1-x)^2.
E.g.f.: 3*(exp(x)*(10*x - 9) + 9).
a(n) = 3*A017281(n-1) = A139280(n)/3.
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

A139249 a(n) = 30*n - 24.

Original entry on oeis.org

6, 36, 66, 96, 126, 156, 186, 216, 246, 276, 306, 336, 366, 396, 426, 456, 486, 516, 546, 576, 606, 636, 666, 696, 726, 756, 786, 816, 846, 876, 906, 936, 966, 996, 1026, 1056, 1086, 1116, 1146, 1176, 1206, 1236, 1266, 1296, 1326, 1356, 1386, 1416, 1446, 1476

Offset: 1

Odimar Fabeny, Jun 06 2008, Jun 07 2008

Multiples of 6 with unit digit equal to 6.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139264, A139279 and A139280. - Reinhard Zumkeller, Jun 22 2008

Cf. A016861.

[30*n - 24: n in [1..60]]; // Vincenzo Librandi, Jun 19 2011

Range[6,7000,30] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2011 *)
NestList[30+#&,6,50] (* or *) LinearRecurrence[{2,-1},{6,36},50] (* Harvey P. Dale, May 27 2018 *)

a(n)=30*n-24 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 30.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 6*x*(1+4*x)/(1-x)^2.
E.g.f.: 6*(exp(x)*(5*x - 4) + 4).
a(n) = 6*A016861(n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

Edited by R. J. Mathar, Jul 20 2008

A139264 a(n) = 70*n - 63.

Original entry on oeis.org

7, 77, 147, 217, 287, 357, 427, 497, 567, 637, 707, 777, 847, 917, 987, 1057, 1127, 1197, 1267, 1337, 1407, 1477, 1547, 1617, 1687, 1757, 1827, 1897, 1967, 2037, 2107, 2177, 2247, 2317, 2387, 2457, 2527, 2597, 2667, 2737, 2807, 2877, 2947, 3017, 3087, 3157, 3227

Offset: 1

Odimar Fabeny, Jun 06 2008

Multiples of 7 with unit digit equal to 7.

Vincenzo Librandi, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Subsequence of A034709, together with A017281, A017293, A139222, A139245, A017329, A139249, A139279 and A139280.

[70*n-63: n in [1..50]]; // Vincenzo Librandi, Jun 19 2011

70*Range[50]-63 (* Harvey P. Dale, May 06 2019 *)

a(n)=70*n-63 \\ Charles R Greathouse IV, Aug 25 2014

a(n) = a(n-1) + 70.
From Elmo R. Oliveira, Apr 04 2025: (Start)
G.f.: 7*x*(1+9*x)/(1-x)^2.
E.g.f.: 7*(exp(x)*(10*x - 9) + 9).
a(n) = 7*A017281(n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

More terms from Reinhard Zumkeller, Jun 22 2008

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

A178158 Numbers n that are divisible by every suffix of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21, 22, 24, 25, 31, 32, 33, 35, 36, 41, 42, 44, 45, 48, 51, 52, 55, 61, 62, 63, 64, 65, 66, 71, 72, 75, 77, 81, 82, 84, 85, 88, 91, 92, 93, 95, 96, 99, 101, 102, 104, 105, 125, 201, 202, 204, 205, 208, 225, 301, 302, 303, 304, 305, 306, 312, 315, 325, 375, 401, 402, 404, 405, 408, 425, 501, 502, 504, 505, 525, 601, 602, 603, 604

Offset: 1

Michel Lagneau, Dec 17 2010

If n = y*10^d+z is in the sequence, where 1<=y<=9 and z < 10^d, then z | y*10^d. - Robert Israel, Oct 17 2018

			9375 is in the sequence because :
.     5 | 9375 ;
.    75 | 9375 ;
.   375 | 9375 ;
.  9375 | 9375 .

Robert Israel, Table of n, a(n) for n = 1..10000 (first 788 terms from Reinhard Zumkeller)
Math Overflow, The number of numbers no greater than n that are divisible by all their suffixes

Cf. A034709 .

Cf. A067251.

import Data.List (tails)
a178158 n = a178158_list !! (n-1)
a178158_list = filter (\suff -> all ((== 0) . (mod suff))
   (map read $ tail $ init $ tails $ show suff :: [Integer])) a067251_list
-- Reinhard Zumkeller, Mar 26 2012

with(numtheory):T:=array(1..5):for n from 1 to 10000 do:ind:=0:l:=length(n):n0:=n:s:=0:for m from 0 to l-1 do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v : s:=s + u*10 ^m:if irem(n,10)<>0 and irem(n, s)=0 then ind:=ind+1:else fi:od:if ind=l then printf(`%d,`, n):else fi:od:
# Alternative:
filter:= proc(x)
  if x mod 10 = 0 then return false fi;
  andmap(t -> type(x/(x mod 10^t),integer), [$1..ilog10(x)])
end proc:
Res:= $1..9:
for d from 1 to 6 do
  for y from 1 to 9 do
    for z in sort(convert(select(`<`,numtheory:-divisors(y*10^d),10^d),list)) do
      if filter(y*10^d+z) then
         Res:= Res, y*10^d+z;
      fi
od od od:
Res; # Robert Israel, Oct 17 2018

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

cp's OEIS Frontend

A038770 Numbers divisible by at least one of their digits.

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A115569 Lynch-Bell numbers: numbers n such that the digits are all different (and do not include 0) and n is divisible by each of its individual digits.

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A139222 a(n) = 30*n - 27.

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A139249 a(n) = 30*n - 24.

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Magma

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A139264 a(n) = 70*n - 63.

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

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