A038770 -id:A038770 - cp's OEIS frontend

A327560 The number of integers m in the range 0 < m < 10^n which are divisible by one or more of their own digits (A038770).

9, 68, 708, 7578, 79952, 832506, 8585583, 87944417, 896452992, 9104962748, 92222435013, 932113080563, 9405187219507, 94771322677210, 953907792350911, 9592689086414459, 96392955785210896, 967997194404428275, 9715595409791983073

Offset: 1

Kevin Ryde, Sep 16 2019

The integers m counted are A038770, so A038770(a(n)) = 10^n-1 is the last of n digits, and A038770(a(n)+1) = 10^n is the first of n+1 digits, for n>=1.
The digit divisibility condition is a regular language so a(n) is a linear recurrence.  Working through a state machine for A038770 shows the recurrence is order 984, though its characteristic polynomial factorizes over rationals into terms of orders at most 36.  The recurrence begins at a(4)..a(987) giving a(988).  See the links for coefficients and generating function.
The biggest root (by magnitude) of the recurrence characteristic polynomial is 10 and its g.f. coefficient is 1 which shows a(n) -> 10^n.  Or simply the number of m containing at least one digit 1 (a subset of those counted here) approaches 10^n per A016189.

Kevin Ryde, Table of n, a(n) for n = 1..999
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 984.

Cf. A038770 (digit strings), A327561 (opposite counts).

Accumulate@ Array[Count[Range[10^(# - 1), 10^# - 1], ?(MemberQ[Divisible[#, Cases[IntegerDigits[#], Except[0]]], True] &)] &, 7] (* _Michael De Vlieger, Sep 30 2019, after Harvey P. Dale at A038770 *)

a(n) = 10^n-1 - A327561(n).

A176659 Partial sums of A038770.

Original entry on oeis.org

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 277, 302, 328, 356, 386, 417, 449, 482, 517, 553, 592, 632, 673, 715, 759, 804, 852, 902, 953, 1005, 1060, 1120, 1181, 1243, 1306, 1370, 1435, 1501, 1571, 1642

Offset: 1

Jonathan Vos Post, Apr 23 2010

Partial sums of numbers divisible by at least one of their digits. Identical to triangular numbers A000217 until a(23), then differs, because 23 is the smallest natural number in the complement of A038770 (A038772, i.e., not divisible by at least one of its digits). Hence this partial sum is the triangular numbers minus the partial sums of A038772, properly offset. The subsequence of primes (of course 3 is the largest prime triangular number) in the partial sum begins: 3, 277, 449, 673, 953, 1181, 1571, 1789, 2027. What are the equivalents in bases other than 10?

			a(23) = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 24 = 277 is prime.

Cf. A038770, A034709, A034837, A038769, A038772.

Accumulate[Select[Range[110],MemberQ[Divisible[#,Cases[ IntegerDigits[ #], Except[ 0]]], True]&]] (* Harvey P. Dale, May 11 2017 *)

a(n) = Sum_{i=1..n} A038770(i).

A038772 Numbers not divisible by any of their digits.

Original entry on oeis.org

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

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 *)

A055238 Numbers that are divisible by at least one of their digits in every base.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 36, 42, 48, 60, 120

Offset: 1

Henry Bottomley, May 04 2000

This list is probably finite and complete. For example 180 fails at base 23 and 240 fails at bases 21, 31 and 33

			9 is included because it can be written as 111111111,1001,100,21,14,13,12,11,10 or 9 and in every case there is a digit which divides 9; 11 is not on the list because in base 4 it is written 23 and 11 is not divisible by either 2 or 3.

Cf. A038770, A055239, A055240, A055241, A055242.

divQ[n_, base_] := AnyTrue[Select[IntegerDigits[n, base], Positive], Divisible[n, #]&];
okQ[n_] := AllTrue[Range[2, n-1], divQ[n, #]&];
Select[Range[1000], okQ] (* Jean-François Alcover, Nov 07 2019 *)

A139138 Numbers divisible by at least two of their digits.

Original entry on oeis.org

11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 131, 132, 135, 138, 140, 141, 142, 144, 145, 147, 148, 150, 151, 152, 153, 155, 156, 161, 162

Offset: 1

Jonathan Vos Post, Jun 05 2008

Digits need not be distinct. This may be considered row 2 of an infinite array whose 1st row is A038770. Each such row is a subset of the ones above it.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Alvin Hoover Belt)
Index entries for 10-automatic sequences

Cf. A038770, A187516.

fQ[n_] := Count[Mod[n, Flatten[IntegerDigits[n] /. {0 -> {}}]], 0] > 1; Select[ Range@ 170, fQ] (* Robert G. Wilson v, Jun 23 2014 *)
Select[Range[200],Count[Divisible[#,Select[IntegerDigits[#], #>0&]], True]>1&] (* Harvey P. Dale, Dec 16 2015 *)

from sympy import factorint
def ok(n): return sum(1 for d in map(int, str(n)) if d > 0 and n%d == 0) > 1
print([k for k in range(163) if ok(k)]) # Michael S. Branicky, Nov 12 2021

More terms from Alvin Hoover Belt, Apr 06 2009

Own omission (140) fixed by Alvin Hoover Belt, Apr 18 2009

A249766 Numbers n = concat(s,t) such that s and t divide n.

Original entry on oeis.org

11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 101, 102, 104, 105, 110, 120, 125, 150, 202, 204, 208, 210, 220, 240, 250, 303, 306, 312, 315, 330, 360, 375, 404, 408, 416, 420, 440, 480, 505, 510, 520, 525, 550, 606, 612, 624, 630, 660, 707, 714, 728

Offset: 1

Paolo P. Lava, Nov 05 2014

Subset of A038770 and A139138.

			306 = concat(3, 06) and 306 / 3 = 102 and 306 / 6 = 51.
76152 = concat(76, 152) and 76152 / 76 = 1002 and 76152 / 152 = 501.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A038770, A139138.

with(numtheory):P:=proc(q) local a,b,k,n;
for n from 1 to q do for k from 1 to ilog10(n) do a:=n mod 10^k; b:=trunc(n/10^k);
if a*b>0 then if type(n/a,integer) and type(n/b,integer) then print(n); break;
fi; fi; od; od; end: P(10^6);

A327561 The number of integers m in the range 0 < m < 10^n which are not divisible by any of their own digits (A038772).

Original entry on oeis.org

0, 31, 291, 2421, 20047, 167493, 1414416, 12055582, 103547007, 895037251, 7777564986, 67886919436, 594812780492, 5228677322789, 46092207649088, 407310913585540, 3607044214789103, 32002805595571724, 284404590208016926

Offset: 1

Kevin Ryde, Sep 19 2019

The integers m counted are A038772 so that A038772(a(n)) is the last there of n digits and A038772(a(n)+1) is the first there of n+1 digits, for n>=2.
The digit divisibility condition is a regular language so a(n) is a linear recurrence.  Working through a state machine for A038772 (or its complement A038770) shows the recurrence is order 983, though its characteristic polynomial factorizes over rationals into terms of orders at most 36.  The recurrence begins at a(4..986) giving a(987).  See the links for recurrence coefficients and generating function.
The biggest root (by magnitude) of the characteristic polynomial is 9 and its g.f. coefficient is 4/21 which shows a(n) -> (4/21)*9^n.

Kevin Ryde, Table of n, a(n) for n = 1..1047
Kevin Ryde, Linear recurrence coefficients and generating function, in a PARI/GP script
Index entries for linear recurrences with constant coefficients, order 983.

Cf. A038770, A038772, A327560.

a(n) = 10^n-1 - A327560(n).

A357929 Numbers that share a (decimal) digit with at least 1 of their proper divisors.

Original entry on oeis.org

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, 74, 75, 77, 80, 81, 82, 84, 85, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 101, 102, 103

Offset: 1

Wesley Ivan Hurt, Oct 21 2022

			74 is in the sequence since it shares a (decimal) digit with at least one of its proper divisors, namely 37. Both numbers share the digit 7.

Wikipedia, Table of divisors

Cf. A038770.

A188454 Numbers n whose decimal digits are distinct and no digit divides n.

Original entry on oeis.org

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, 239, 247, 249, 253, 257, 259, 263, 267, 269, 283, 289, 293, 307, 308, 329, 346, 347, 349, 356, 358, 359, 367, 370, 374

Offset: 1

Andy Edwards, Mar 31 2011

These may not contain 1 or have a 2 or 5 as the last digit. They include prime numbers not containing the digit 1 and composites with a smallest prime factor > 10 and obeying the other constraints (e.g. the largest case is 987654203 = 31*31859813).
The first even case is 34. The first consecutive pair is {37, 38}. {56,57,58,59} is a consecutive quadruple which is the maximal size for such a subset.
There are 202623 terms in this sequence. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A038772, A034709, A034837, A038769, A038770.

dddQ[n_]:=Module[{dcn=DigitCount[n]},Max[dcn]==1&&First[dcn]==0 && Union[ Divisible[n,Select[IntegerDigits[n],#!=0&]]]=={False}]; Select[Range[ 400],dddQ] (* Harvey P. Dale, May 01 2012 *)

cp's OEIS Frontend

A327560 The number of integers m in the range 0 < m < 10^n which are divisible by one or more of their own digits (A038770).

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A176659 Partial sums of A038770.

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A038772 Numbers not divisible by any of their digits.

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

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A055238 Numbers that are divisible by at least one of their digits in every base.

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A139138 Numbers divisible by at least two of their digits.

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A249766 Numbers n = concat(s,t) such that s and t divide n.

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A327561 The number of integers m in the range 0 < m < 10^n which are not divisible by any of their own digits (A038772).