A034838 -id:A034838 - cp's OEIS frontend

A225299 Numbers divisible by the square of each digit.

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))

A285271 Numbers that are divisible by each of their digits but that are either not divisible by the sum of their digits or are not divisible by the product of their digits or both.

Original entry on oeis.org

11, 15, 22, 33, 44, 48, 55, 66, 77, 88, 99, 115, 122, 124, 126, 128, 155, 162, 168, 175, 184, 212, 222, 244, 248, 264, 288, 324, 333, 336, 366, 384, 396, 412, 424, 444, 448, 488, 515, 555, 636, 648, 666, 672, 728, 777, 784, 816, 824, 848, 864, 888, 936, 999, 1111, 1112

Offset: 1

Bernard Schott, Jun 24 2017

Numbers which are divisible by the sum and the product of their digits (A038186) are also divisible by each of their digits (A034838)

The product of the digits of n are trivially divisible by each digit; so if that product divides n, each digit must divide n. - Franklin T. Adams-Watters, Jul 02 2017

			15 is divisible by its digits 1 and 5, and 15 is divisible by the product of its digits 1*5 = 5, but 15 is not divisible by the sum of its digits 1+5 = 6, hence 15 is a term.
48 is divisible by its digits 4 and 8, and 48 is divisible by the sum of its digits 4+8 = 12, but 48 is not divisible by the product of its digits 4*8 = 32, hence 48 is a term.
124 is divisible by its digits 1, 2 and 4, but 124 is not divisible by the product of its digits 1*2*4 = 8 and 124 is not divisible by the sum of its digits 1+2+4 = 7, hence 124 is a term.
24 is divisible by its digits 2 and 4, and 24 is divisible by the sum of its digits 2+4 = 6, and 24 is also divisible by the product of its digits 2*4 = 8, hence 24 is NOT a term.

Robert G. Wilson v, Table of n, a(n) for n = 1..1520

Subsequence of A034838.

Cf. A005349, A007602, A038186.

filter:= proc(n) local F;
   F:= convert(n,base,10);
   andmap(t -> t > 0 and n mod t = 0, F) and not(n mod convert(F,`+`) = 0 and n mod convert(F,`*`) = 0)
end proc:
select(filter, [$11 .. 2000]); # Robert Israel, Jul 05 2017

fQ[n_] := Block[{ind = IntegerDigits@ n}, Union[ IntegerQ@# & /@ (n/ind)] == {True} && (!IntegerQ[n/Plus @@ ind] || !IntegerQ[n/Times @@ ind])]; Select[Range@ 1112, fQ] (* Robert G. Wilson v, Jul 05 2017 *)
nddQ[n_]:=With[{idn=IntegerDigits[n]},FreeQ[idn,0]&&AllTrue[n/idn,IntegerQ]&&(!IntegerQ[n/Times@@idn]||!IntegerQ[n/Total[idn]])]; Select[Range[1200],nddQ] (* Harvey P. Dale, May 04 2025 *)

isok(n) = {d = digits(n); if (vecmin(d), for (k=1, #d, if (n % d[k], return (0));); return ((n % vecsum(d)) || (n % prod(k=1, #d, d[k])));); return (0);} \\ Michel Marcus, Jul 02 2017

Definition clarified by Harvey P. Dale, May 04 2025

A330981 Remodd numbers: having an odd remainder modulo all of their digits, digit 0 forbidden.

Original entry on oeis.org

43, 47, 73, 87, 223, 227, 253, 267, 283, 289, 337, 343, 349, 367, 379, 397, 433, 439, 463, 467, 469, 477, 489, 493, 523, 553, 583, 643, 647, 649, 669, 673, 677, 687, 689, 733, 747, 787, 799, 823, 827, 829, 849, 853, 869, 883, 887, 889, 943, 997

Offset: 1

Eric Angelini and M. F. Hasler, Jan 05 2020

No term may have a digit 0 or 1, therefore the asymptotic density is zero and would be so even if the definition is changed to "digits 0 are allowed but ignored", since pandigital numbers A171102 have asymptotic density 1.
Does not contain any remeven number (A330982), thus in particular none of A010785 (repdigits) or its superset A034838 (divisible by all digits) or A014263 (only even digits). Also no multiples of 2 or 5 (A005843 or A008587) which are even modulo the last digit (unless it is 0), so all terms end in 3, 7 or 9.
Contains the infinite subsequence (43, 433, 4333, ...), but after {47, 477, 4777} not 47777 = 6825*7 + 2, and after {73, 733} not 7333 = 1047*7 + 4, and after {87, 887} not 8887 = 1269*7 + 4.
The first term which contains the digits 2..9 is a(784795) = 224567983. - Giovanni Resta, Jan 07 2020

			43 is in the sequence because 43 mod 4 = 3 and 43 mod 3 = 1 both are odd.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Angelini, Remeven numbers, SeqFan list, Jan 05 2020.

Cf. A330982 (remeven numbers).

Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).

[k:k in [1..1000]|not 0 in Intseq(k) and forall{d:d in Intseq(k)|IsOdd(k mod d)}]; // Marius A. Burtea, Jan 07 2020

select( {is(n, d=Set(digits(n)))=d[1]&&!for(j=1,#d, bittest(n%d[j],0)||return)}, [1..2000])

A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 22, 24, 26, 28, 32, 33, 35, 36, 38, 42, 44, 46, 48, 52, 54, 55, 62, 64, 66, 68, 72, 74, 76, 77, 82, 84, 86, 88, 92, 94, 96, 98, 99, 111, 112, 113, 114, 115, 116, 118, 119, 122, 124, 126, 128, 131, 132, 134, 135, 137, 138, 142, 144, 146, 148, 152

Offset: 1

Eric Angelini and M. F. Hasler, Jan 05 2020

The sequence is a subset of the zeroless numbers A052382 which have asymptotic density 0 because they are in the complement of pandigital numbers A171102 which have asymptotic density 1. But does it have finite density within A052382?
It contains all repdigit numbers A010785 \ {0} and also all numbers with only even digits A014263 \ {0} and all numbers divisible by all of their digits, A034838.
The graph is self-similar, it looks the same whether we take the graph of values < 10^4 or that of values < 10^5 etc.: In the range 0 < a(n) < 10^(k+1), there are jumps of size > 10^k/9 where the values cross the limits d*10^k, 1 <= d <= 9 (from a(n) <= {d-1}9...9 to a(n+1) >= d1...1, since 0's are forbidden).
There are N = (0, 9, 48, 303, 2190, 15871, 119442, 930324, ...) terms below 10^k, k >= 0; these N(k) are also the indices of terms a(N(k)) = 10^k-1 (k>0), which are followed by repunits a(N(k)+1) = a(N(k+1))/9 (k >= 0).
The smallest zeroless pandigital term is a(8455060) = 123567894. - Giovanni Resta, Jan 08 2020

			12 is in the sequence because 12 % 1 = 0 and 12 % 2 = 0 both are even, where x % y is the remainder of x divided by y.
13 is not in the sequence because 13 % 3 = 1 is odd.

M. F. Hasler, Table of n, a(n) for n = 1..10000, Jan 07 2020.
Eric Angelini, Remeven numbers, SeqFan list, Jan 05 2020.

Cf. A330981 (remodd numbers).

Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).

[k:k in [1..160]|not 0 in Intseq(k) and forall{d:d in Intseq(k)|IsEven(k mod d)}]; // Marius A. Burtea, Jan 08 2020

Select[Range[200],DigitCount[#,10,0]==0&&AllTrue[Mod[#,IntegerDigits[ #]],EvenQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 02 2020 *)

select( {is_A330982(n,d=digits(n))=vecmin(d)&&!for(j=1,#d, bittest(n%d[j],0)&&return)}, [1..200]) \\ Using Set(digits) is about 20% slower

A342650 Numbers divisible both by their nonzero individual digits and by the sum of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 24, 30, 36, 40, 48, 50, 60, 70, 80, 90, 100, 102, 110, 111, 112, 120, 126, 132, 135, 140, 144, 150, 162, 200, 204, 210, 216, 220, 222, 224, 240, 264, 280, 288, 300, 306, 312, 315, 324, 330, 333, 336, 360, 396, 400, 408, 420, 432, 440, 444, 448, 480, 500

Offset: 1

Bernard Schott, Mar 18 2021

Equivalently, Niven numbers that are divisible by their nonzero digits. A Niven number (A005349) is a number that is divisible by the sum of its digits.
Niven numbers without zero digit that are divisible by their individual digits are in A051004.
Differs from super Niven numbers, the first 25 terms are the same, then A328273(26) = 120 while a(26) = 111.
This sequence is infinite since if m is a term, then 10*m is another term.

			102 is divisible by its nonzero digits 1 and 2, and 102 is also divisible by the sum of its digits 1 + 0 + 2 = 3, then 102 is a term.

Intersection of A002796 and A005349.
Supersequence of A051004.
Cf. A034838, A328273.

q[n_] := AllTrue[(d = IntegerDigits[n]), # == 0 || Divisible[n, #] &] && Divisible[n, Plus @@ d]; Select[Range[500], q] (* Amiram Eldar, Mar 18 2021 *)

isok(m) = if (!(m % sumdigits(m)), my(d=select(x->(x>0), Set(digits(m)))); setintersect(d, divisors(m)) == d); \\ Michel Marcus, Mar 18 2021

A346267 Number of n-digit integers that are divisible by all their digits.

Original entry on oeis.org

9, 14, 56, 260, 1306, 7394, 43951, 273509, 1761231, 11635311, 78551945, 539622083, 3762656337, 26579694095, 189938085415, 1371475597978, 9996841746666, 73499537666630, 544684691301292, 4065992493282511, 30555869899381064, 231043525054841279, 1756887541883726014

Offset: 1

Michel Marcus, Jul 12 2021

From Michael S. Branicky, Jul 13 2021: (Start)
For a(12), the count for 12-digit numbers ending in 1..9 is 89385, 126484057, 89966, 152213988, 1354818, 127833463, 72297, 131400895, 83214, resp.
Terms can be computed using reachability analysis (see program in links) on the following finite automaton with 315906 reachable states: Di = {0, ..., i-1}, D = {1, ..., 9}; P(A) denotes the power set of A; Z the empty set; U, union; Q = D2 X ... X D9 X P({2, ..., 9}), Sigma = D, s = (0, ..., 0) X Z; delta((q2, ..., q9; A), c) = (10*q2+c mod 2, ..., 10*q9+c mod 9; A'), where A' = A if c = 1 and A U c otherwise; F = {q X A | qi = 0 for i in A}.
Alternatively, the following smaller finite automaton may similarly be analyzed (see alternate program in links) to compute sequence terms: Q = {(r, m) = (remainder-so-far modulo 2520, lcm(seen digits))}; Sigma = {0, ..., 9}; s = (0, 1); F = {(r, m) | r mod m == 0}; delta((r, m), c) = (10*q+c mod 2520, lcm(r, c)) for c <> 0, delta(q, 0) dies for all q. (End)

			In A034838, we have (1, 2, 3, 4, 5, 6, 7, 8, 9) so a(1) = 9.
And we have (11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99) so a(2) = 14.

Michael S. Branicky, Table of n, a(n) for n = 1..1110 (terms <= 1000 digits)
Michael S. Branicky, Python program
Michael S. Branicky, Python program based on finite automaton
Michael S. Branicky, Python program based on alternate finite automaton
Michael S. Branicky, Terms 1..5000 for discovering linear recurrence
Ana Rechtman, Juillet 2021, 2e défi, Images des Mathématiques, CNRS, 2021 (in French).
Kevin Ryde, Linear recurrence coefficients and generating function, in a PARI/GP script
Kevin Ryde, Perl program making a state machine with Foma to verify the linear recurrence
Index entries for linear recurrences with constant coefficients, order 928.

Cf. A034838.

is(n)=my(d=Set(digits(n))); d[1]&&!forstep(i=#d, 1, -1, n%d[i]&&return); \\ A034838
a(n) = sum(k=10^(n-1), 10^n-1, is(k));

# see links for a faster version and FA-based programs
def ok(n): return all(d != '0' and n%int(d) == 0 for d in set(str(n)))
def a(n): return sum(ok(k) for k in range(10**(n-1), 10**n))
print([a(n) for n in range(1, 7)]) # Michael S. Branicky, Jul 12 2021

a(9) and beyond from Michael S. Branicky, Jul 13 2021

A082937 Palindromes divisible by each of their digits.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 212, 222, 333, 424, 444, 515, 555, 636, 666, 777, 848, 888, 999, 1111, 1771, 2112, 2222, 2772, 3333, 4224, 4444, 5115, 5555, 5775, 6336, 6666, 7777, 8448, 8888, 9999, 11111, 11711, 13131, 17171

Offset: 1

Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 24 2003

			424 is divisible by both 4 and 2.

Harvey P. Dale, Table of n, a(n) for n = 1..300

Intersection of A002113 (palindromes in base 10) and A034838.

Subsequence of A052382 (zeroless numbers).

Select[Range[20000],PalindromeQ[#]&&FreeQ[IntegerDigits[#],0]&&Union[Mod[ #,IntegerDigits[ #]]]=={0}&] (* Harvey P. Dale, Feb 08 2024 *)

isok(n) = {my(d=digits(n)); if ((d == Vecrev(d)) && vecmin(d), d = Set(d); for (i=1, #d, if (n % d[i], return(0));); return (1);); return (0);} \\ Michel Marcus, May 26 2019

More terms from Michel Marcus, May 26 2019

Definition clarified by Harvey P. Dale, Feb 08 2024

A093785 Numbers that are divisible by every digit in their Roman numeral representation.

Original entry on oeis.org

1, 2, 3, 5, 10, 20, 30, 50, 100, 200, 300, 500, 1000, 2000, 3000

Offset: 1

Reinhard Zumkeller, May 17 2004

The sequence contains only values less than 4000, see A078715 for a discussion on the Roman 4M-problem.

			I, II, III, V, X, XX, XXX, L, C, CC, CCC, D, M, MM, MMM.

Stephanus Gibbs, Roman Numeral and Date Conversion
Gerard Schildberger, The first 3999 numbers in Roman numerals
Eric Weisstein's World of Mathematics, Roman Numerals
Wikipedia, Roman numerals

Cf. A034838.

Cf. A061493.

a093785 n = a093785_list !! (n-1)
a093785_list = filter p [1..3999] where
   p v = q $ a061493 v where
     q w = w == 0 || v `mod` ([0,1,5,10,50,100,500,1000] !! d') == 0 && q w'
          where  (w',d) = divMod w 10; d' = fromInteger d
-- Reinhard Zumkeller, Apr 14 2013

A182183 Numbers k such that the divisors of k 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, 55, 66, 77, 88, 99, 132, 264, 1111111111111111111, 2222222222222222222, 3333333333333333333, 4444444444444444444, 5555555555555555555, 6666666666666666666, 7777777777777777777, 8888888888888888888

Offset: 1

Jaroslav Krizek, Apr 17 2012

Subsequence of A209933 (numbers that are divisible by all digits of their divisors).
All divisors of numbers in this sequence are also in the sequence.
The primitive elements of this sequence are A116692. No member of this sequence is divisible by a prime outside this sequence. - Charles R Greathouse IV, Apr 17 2012

			Number 48 with divisors 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 is not in the sequence because 6 is not a divisor of 16.

Subsequence of A034838. A116692 is a subsequence.
Cf. A209933.
Cf. A027751.
Cf. A027751.

import Data.List ((\\))
a182183 n = a182183_list !! (n-1)
a182183_list = f a209933_list [1] where
   f (x:xs) ys =
     if null (a027751_row x \\ ys) then x : f xs (x : ys) else f xs ys
-- Reinhard Zumkeller, Apr 19 2012

all(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
is(n)=fordiv(n,d,if(!all(d),return(0)));1 \\ Charles R Greathouse IV, Apr 17 2012

a(24)-a(31) from Charles R Greathouse IV, Apr 17 2012

A331342 Lexicographically earliest sequence of distinct terms a(n) indivisible by all of their digits that become divisible by all of their digits when a(n+1) is added to a(n).

Original entry on oeis.org

23, 43, 34, 54, 57, 58, 53, 46, 69, 59, 29, 37, 74, 38, 73, 49, 79, 47, 68, 56, 76, 86, 89, 223, 389, 247, 377, 67, 257, 367, 269, 97, 27, 397, 227, 439, 233, 379, 293, 343, 323, 289, 347, 277, 359, 253, 83, 229, 259, 353, 283, 329, 337, 87, 249, 239, 94, 338, 334, 78, 346, 98, 457, 479, 634, 477, 638

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Jan 14 2020

Is this sequence a reordering of A038772?

			a(1) = 23 is not divisible by 2 and not divisible by 3. When a(2) = 43 is added to a(1) = 23, the result (66) is divisible by all its digits.
a(2) = 43 is not divisible by 4 and not divisible by 3. When a(3) = 34 is added to a(2) = 43, the result (77) is divisible by all its digits.
a(3) = 34 is not divisible by 3 and not divisible by 4. When a(4) = 54 is added to a(3) = 34, the result (88) is divisible by all its digits.
a(4) = 54 is not divisible by 5 and not divisible by 4. When a(5) = 57 is added to a(4) = 54, the result (111) is divisible by all its digits.
a(5) = 57 is not divisible by 5 and not divisible by 7. When a(6) = 58 is added to a(5) = 57, the result (115) is divisible by all its digits....

Jean-Marc Falcoz, Table of n, a(n) for n = 1..20001

Cf. A038772, A034838.

cp's OEIS Frontend

A225299 Numbers divisible by the square of each digit.

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A285271 Numbers that are divisible by each of their digits but that are either not divisible by the sum of their digits or are not divisible by the product of their digits or both.

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A330981 Remodd numbers: having an odd remainder modulo all of their digits, digit 0 forbidden.

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Magma

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A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.

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Magma

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PARI

A342650 Numbers divisible both by their nonzero individual digits and by the sum of their digits.

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Mathematica

PARI

A346267 Number of n-digit integers that are divisible by all their digits.

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PARI

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A082937 Palindromes divisible by each of their digits.

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