A038769 -id:A038769 - 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

A330348 a(n) is the number of divisors of n whose last digit equals the last digit 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, 2, 2, 2, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 2, 2, 1, 2, 3, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 4, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 4, 1, 2, 1, 1, 4, 2, 2, 1, 3, 2, 1, 1, 2, 1

Offset: 1

Bernard Schott, Jun 04 2020

Inspired by Project Euler, Problem 474 (see link).
a(n) >= 1.
When n > 10 ends with 0, 1, 2 or 5, then a(n) >= 2.
The first 19 terms are the same as A038769, but a(20) = 2 and A038769(20) = 1.
From Robert Israel, Jun 04 2020: (Start)
a(10*n) = A000005(n).
If n is odd, then a(2*n) = a(n) and a(5*n) = A000005(n). (End)
Integers all of whose divisors end with the same last digit (which is necessarily 1) are in A004615. - Bernard Schott, May 07 2021

			The divisors of 12 that end in 2 are 2 and 12, so a(12) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000
Project Euler, Problem 474: Last digits of divisors

Cf. A000005 (number of divisors), A004615, A010879 (last digit of n), A038769.

f:= proc(n) local t;
t:= n mod 10;
nops(select(k -> k mod 10 = t, numtheory:-divisors(n)))
end proc:
map(f, [$1..100]); # Robert Israel, Jun 04 2020

a[n_] := DivisorSum[n, 1 &, Mod[# - n, 10] == 0 &]; Array[a, 100] (* Amiram Eldar, Jun 04 2020 *)

a(n) = my(u=n%10); sumdiv(n, d, d%10 == u); \\ Michel Marcus, Jun 04 2020

from sympy import divisors
def a(n): return sum((n-d)%10 == 0 for d in divisors(n, generator=True))
print([a(n) for n in range(1, 90)]) # Michael S. Branicky, Aug 15 2022

A055639 Number of nonzero digits of n which are not factors of n.

Original entry on oeis.org

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

Offset: 1

Henry Bottomley, Jun 06 2000

Marius A. Burtea, Table of n, a(n) for n = 1..5000

Cf. A002796, A038769.

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

Array[Count[Position[Most@ DigitCount@ #, ?(# > 0 &)][[All, 1]], k /; Mod[#, k] != 0] &, 105] (* Michael De Vlieger, Dec 23 2019 *)

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).

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 *)

A176660 Partial sums of A038772.

Original entry on oeis.org

23, 50, 79, 113, 150, 188, 231, 277, 324, 373, 426, 480, 536, 593, 651, 710, 777, 845, 914, 987, 1061, 1137, 1215, 1294, 1377, 1463, 1550, 1639, 1733, 1830, 1928, 2131, 2338, 2547, 2770, 2997, 3226, 3459, 3698, 3945, 4194, 4447, 4704, 4963, 5226, 5493

Offset: 1

Jonathan Vos Post, Apr 23 2010

Partial sums of numbers not divisible by any of their digits. The partial sums of the complement (A038770, numbers divisible by at least one of their digits) is A176659. Hence the sum of the two partial sums (properly offset) is triangular numbers A000217. The subsequence of primes in the partial sum begins: 23, 79, 113, 277, 373, 593, 1061, 1733, 2131, 4447, 9479. The smallest square in the sequence is 324.

			a(4) = 23 + 27 + 29 + 34 = 113 is prime.

Cf. A038772, 034709, A034837, A038769, A038770, A176659.

a(n) = SUM[i=1..n] A038772(i).

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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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A330348 a(n) is the number of divisors of n whose last digit equals the last digit of n.

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A055639 Number of nonzero digits of n which are not factors of n.

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A176659 Partial sums of A038770.

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

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A176660 Partial sums of A038772.

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