A011531 -id:A011531 - cp's OEIS frontend

A208270 Primes containing a digit 1.

11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571, 601, 613, 617, 619, 631, 641, 661

Offset: 1

Jaroslav Krizek, Mar 04 2012

Subsequence of A011531, A062634, A092911 and A092912.
Supersequence of A106101, A045707 and A030430.
Complement of A208271 with respect to A011531.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of n

Cf. A208271 (nonprimes containing a digit 1), A011531 (numbers containing a digit 1).

Complement of A038603 in A000040. - M. F. Hasler, Mar 05 2012

[p: p in PrimesUpTo(400) | 1 in Intseq(p)]; // Vincenzo Librandi, Apr 29 2019

Select[Prime[Range[124]], MemberQ[IntegerDigits[#], 1] &](* Jayanta Basu, Apr 01 2013 *)
Select[Prime[Range[200]],DigitCount[#,10,1]>0&] (* Harvey P. Dale, Dec 15 2020 *)

forprime(p=2,1e3,s=vecsort(eval(Vec(Str(p))),,8);if(s[1]==1||(s[1]==0&&s[2]==1),print1(p", "))) \\ Charles R Greathouse IV, Mar 04 2012

is_A208270(n)=isprime(n)&setsearch(Set(Vec(Str(n))),1) \\ M. F. Hasler, Mar 05 2012

a(n) ~ n log n since the sequence contains almost all primes. - Charles R Greathouse IV, Mar 04 2012

A293873 Numbers having '13' as substring of their digits.

Original entry on oeis.org

13, 113, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 213, 313, 413, 513, 613, 713, 813, 913, 1013, 1113, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1213, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312, 1313, 1314, 1315

Offset: 1

M. F. Hasler, Oct 18 2017

Row 13 of A292690 and A293869. A121033 is the subsequence of multiples of 13.

Index entries for 10-automatic sequences

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293874, A293875, A293876, A293877, A293878, A293879, A293880: same for '10' - '20'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[1350],SequenceCount[IntegerDigits[#],{1,3}]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 31 2017 *)

is_A293873 = has(n, p=13, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

a(n) ~ n. - Charles R Greathouse IV, Nov 02 2022

A293880 Numbers having '20' as substring of their digits.

Original entry on oeis.org

20, 120, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 220, 320, 420, 520, 620, 720, 820, 920, 1020, 1120, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1220, 1320, 1420, 1520, 1620, 1720, 1820, 1920, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010

Offset: 1

M. F. Hasler, Oct 18 2017

Row 20 of A292690 and A293869. A121040 lists the terms which are divisible by 19.

Index entries for 10-automatic sequences

Cf. A292690, A293869.
Cf. A011540, A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539: analog for '0' - '9'.
Cf. A293870, A293871, A293872, A293873, A293874, A293875, A293876, A293877, A293878, A293879: same for '10' - '19'.
Cf. A121041, A121022, A121023, A121024, A121025, A121026, A121027, A121028, A121029, A121030, A121031, A121032, A121033, A121034, A121035, A121036, A121037, A121038, A121039, A121040: subsequences of the above, containing only multiples of the pattern p.

Select[Range[2100],SequenceCount[IntegerDigits[#],{2,0}]>0&] (* Harvey P. Dale, Jul 25 2021 *)

is_A293880 = has(n, p=20, m=10^#Str(p))=until(p>n\=10, n%m==p&&return(1))

a(n) ~ n. - Charles R Greathouse IV, Nov 02 2022

A062634 Numbers k such that every divisor of k contains the digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 221, 241, 251, 271, 281, 311, 313, 317, 331, 341, 361, 401, 419, 421, 431, 451, 461, 491, 521

Offset: 1

Erich Friedman, Jul 04 2001

First composite term is 121. All powers of 11 are in the sequence. - Alonso del Arte, Sep 29 2013

			143 has divisors 1, 11, 13 and 143, all of which contain the digit 1.

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

Cf. A027750, subsequence of A011531; A206159 and A208270 are subsequences.

Cf. A001020 (powers of 11).

a062634 n = a062634_list !! (n-1)
a062634_list = filter
   (and . map ((elem '1') . show) . a027750_row) a011531_list
-- Reinhard Zumkeller, Feb 05 2012

q:= n-> andmap(x-> 1 in convert(x, base, 10), numtheory[divisors](n)):
select(q, [$1..1000])[];  # Alois P. Heinz, May 09 2022

fQ[n_, dgt_] := Union[ MemberQ[#, dgt] & /@ IntegerDigits@ Rest@ Divisors@ n][[1]]; Select[ Range[2, 525], fQ[#, 1] &] (* Robert G. Wilson v, Jun 11 2014 *)

isok(m) = fordiv(m, d, if (! #select(x->(x==1), digits(d)), return(0))); return(1); \\ Michel Marcus, May 09 2022

Offset corrected by Reinhard Zumkeller, Feb 05 2012

A069715 GCD of digits of n is 1.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 31, 32, 34, 35, 37, 38, 41, 43, 45, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 61, 65, 67, 71, 72, 73, 74, 75, 76, 78, 79, 81, 83, 85, 87, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108

Offset: 1

Labos Elemer, Apr 02 2002

			All numbers with at least one digit equal to 1 are here.

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

Cf. A011531 (subsequence), A240913 (subsequence).

a069715 n = a069715_list !! (n-1)
a069715_list = filter ((== 1) . a052423) [1..]
-- Reinhard Zumkeller, Apr 14 2014

Do[s=Apply[GCD, IntegerDigits[n]]; If[Equal[s, 1], Print[n]], {n, 1, 256}]

is(n)=gcd(digits(n))==1 \\ Charles R Greathouse IV, Nov 01 2014

A052423(a(n)) = 1. - Reinhard Zumkeller, Apr 14 2014

a(n) ~ n. In fact a(n) = n + O(n^(log 5/log 10)). - Charles R Greathouse IV, Nov 01 2014

A121042 Smallest divisor of n that is also contained in the decimal representation of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 23, 2, 5, 2, 27, 2, 29, 3, 1, 2, 3, 34, 5, 3, 37, 38, 3, 4, 1, 2, 43, 4, 5, 46, 47, 4, 49, 5, 1, 2, 53, 54, 5, 56, 57, 58, 59, 6, 1, 2, 3, 4, 5, 6, 67, 68, 69, 7, 1, 2, 73, 74, 5, 76, 7, 78, 79, 8, 1, 2, 83, 4, 5, 86, 87, 8, 89, 9, 1

Offset: 1

Reinhard Zumkeller, Jul 21 2006

1 <= a(n) <= n;
a(A011531(n)) = 1; a(n) = n iff A121041(n) = 1.
a(n) = 1 for almost all n (measure 1). - Charles R Greathouse IV, Mar 31 2016

			a(48) = Min{4, 8, 48} = 4;
a(49) = Min{49} = 49;
a(120) = Min{1, 2, 12, 20, 120} = 1;
a(121) = Min{1} = 1.

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A011531, A027750, A121041, A383749 (fixed points).

A121042[n_] := SelectFirst[Divisors[n], StringContainsQ[IntegerString[n], IntegerString[#]] &];
Array[A121042, 100] (* Paolo Xausa, May 12 2025 *)

substr(a,b)=a=digits(a);b=digits(b); for(i=0,#a-#b, for(j=1,#b, if(a[i+j]!=b[j], next(2))); return(1)); 0
a(n)=fordiv(n,d, if(substr(n,d), return(d))) \\ Charles R Greathouse IV, Mar 31 2016

A175688 Numbers k with property that arithmetic mean of its digits is both an integer and one of the digits of k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 102, 111, 120, 123, 132, 135, 147, 153, 159, 174, 195, 201, 204, 210, 213, 222, 231, 234, 240, 243, 246, 258, 264, 285, 306, 312, 315, 321, 324, 333, 342, 345, 351, 354, 357, 360, 369, 375, 396, 402

Offset: 1

Claudio Meller, Aug 09 2010

Subsequence of A061383.

A180160(a(n)) = 0. - Reinhard Zumkeller, Aug 15 2010

			135 is in the list because (1+3+5)/3 = 3 and 3 is a digit of 135.

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

Cf. A007953, A055642, A004426, A004427; A011531, A011532, A011533, A011534, A011535, A011536, A011537, A011538, A011539.

Cf. A052018. - Reinhard Zumkeller, Aug 15 2010

a175688 n = a175688_list !! (n-1)
a175688_list = filter f [0..] where
   f x = m == 0 && ("0123456789" !! avg) `elem` show x
         where (avg, m) = divMod (a007953 x) (a055642 x)
-- Reinhard Zumkeller, Jun 18 2013

idQ[n_]:=Module[{idn=IntegerDigits[n],m},m=Mean[idn];IntegerQ[m] && MemberQ[idn,m]]; Select[Range[0,500],idQ] (* Harvey P. Dale, Jun 10 2011 *)

Edited by Reinhard Zumkeller, Aug 13 2010

A092911 Numbers all of whose divisors can be formed using their digits. Divisor digits are a subset of the digits of the number.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 241, 251, 271, 281, 311, 313, 317, 331, 401, 419, 421, 431, 461, 491, 521, 541, 571

Offset: 1

Amarnath Murthy, Mar 14 2004

All primes containing 1 are members.

Sequence is a subsequence of A011531. The first nonprime terms of the sequence are 1, 121, 125, 1207, 1255, 1379, 10201, 10379, 11009, 11209, 12419, 12709, 12755, ... - R. J. Mathar, Jul 26 2007

			131 is a term. 143 is not a term as the divisor 11 contains two 1's.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A011531, A062634, A092912.

isA092911 := proc(n) local digs, digsleft,divs, d,i,j ; digs := convert(n,base,10) ; divs := numtheory[divisors](n) ; for i from 1 to nops(divs) do digsleft := digs ; d := convert(op(i,divs),base,10) ; for j in d do if member(j,digsleft,'jposit') then digsleft := subsop(jposit=NULL,digsleft) ; else RETURN(false) ; fi ; od ; od ; RETURN(true) ; end: for n from 1 to 600 do if isA092911(n) then printf("%d, ",n) ; fi ; od ; # R. J. Mathar, Jul 26 2007

subQ[s1_, s2_] := AllTrue[Count[s1, #] & /@ (First /@ (t = Tally[s2])) - Last /@ t, # >= 0 &]; digQ[n1_, n2_] := subQ[IntegerDigits[n1], IntegerDigits[n2]]; seqQ[n_] := AllTrue[Most@Divisors[n], digQ[n, #] &]; Select[Range[600], seqQ] (* Amiram Eldar, Nov 12 2020 *)

from sympy import divisors
from collections import Counter
def ok(n):
  ncounts = Counter(str(n))
  for d in divisors(n)[:-1]:
    divcounts = Counter(str(d))
    if any(ncounts[c] < divcounts[c] for c in divcounts): return False
  return True
print(list(filter(ok, range(1, 630)))) # Michael S. Branicky, May 08 2021

Corrected and extended by R. J. Mathar, Jul 26 2007

A241146 Least number k such that k and n*k share at least one digit.

Original entry on oeis.org

1, 10, 5, 10, 5, 2, 5, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 9, 5, 10, 5, 2, 5, 6, 5, 5, 1, 10, 5, 10, 4, 2, 4, 10, 5, 10, 1, 10, 5, 3, 3, 2, 5, 8, 5, 5, 1, 10, 5, 7, 5, 2, 5, 10, 5, 2, 1, 2, 2, 2, 5, 2, 5, 7, 5, 5, 1, 10, 5, 10, 5, 2, 3, 3, 3, 10, 1, 7, 5, 10, 4, 2, 4, 9, 5, 5, 1, 10, 5, 6, 5, 2, 5, 8, 5, 1

Offset: 1

Robert G. Wilson v, Apr 16 2014

\ 10^3...10^4....10^5.....10^6......10^7.......10^8........10^9
k\
.1..272...3440...40952...468560...5217032...56953280...612579512
.2..149...1613...16837...171325...1710773...16837421...163825573
.3...87...1091...12038...124060...1225493...11762254...110573419
.4...62....710....7196....68280....621670....5502346....47710882
.5..248...1914...14674...111846....848318....6407338....48220222
.6...26....246....2087....16749....129768.....980911.....7280424
.7...36....323....2587....19368....138838.....966609.....6591845
.8...20....156....1095.....7199.....45386.....277985.....1667513
.9...22....162....1028.....6055.....34178.....187661.....1010240
10...78....345....1506.....6558.....28544.....124195......540370
The sequence of numbers whose first digit is k:
.1: 1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 100, …, . A011531?
.2: 6, 26, 36, 46, 56, 60, 62, 63, 64, 66, 76, 86, 96, 206, 226, 236, 246, 256, 260, 262, …, .
.3: 44, 45, 77, 78, 79, 244, 245, 277, 278, 279, 344, 345, 377, 378, 379, 434, 435, 437, 438, …, .
.4: 35, 37, 85, 87, 235, 237, 285, 287, 335, 337, 350, 352, 353, 354, 355, 357, 358, 359, 365, …, .
.5: 3, 5, 7, 9, 23, 25, 27, 29, 30, 33, 39, 43, 47, 49, 50, 53, 55, 57, 59, 65, 67, 69, 70, 73, …, .
.6: 28, 94, 228, 268, 272, 274, 280, 282, 294, 328, 394, 428, 494, 528, 594, 694, 728, 828, 894, …, .
.7: 54, 68, 82, 248, 252, 254, 382, 388, 392, 398, 468, 482, 532, 534, 538, 540, 542, 554, 568, …, .
.8: 48, 98, 232, 234, 298, 348, 480, 484, 498, 548, 598, 698, 732, 734, 748, 848, 898, 980, 984, …, .
.9: 22, 88, 220, 222, 288, 322, 324, 332, 422, 488, 522, 552, 588, 658, 688, 722, 880, 884, 888, …, .
10: 2, 4, 8, 20, 24, 32, 34, 38, 40, 42, 52, 58, 72, 74, 80, 84, 92, 200, 202, 204, 208, 224, …, .

Cf. A129845, A129846, A129848, A241141, A241142, A241143, A241144, A241145.

f[n_] := Block[{k = 1}, While[ Intersection[ IntegerDigits[k], IntegerDigits[n*k]] == {}, k++]; k]; Array[f, 100]

If a(n) = k so does a(10n).

A206159 Numbers needing at most two digits to write all positive divisors in decimal representation.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 22, 31, 33, 41, 55, 61, 71, 77, 101, 113, 121, 131, 151, 181, 191, 199, 211, 311, 313, 331, 661, 811, 881, 911, 919, 991, 1111, 1117, 1151, 1171, 1181, 1511, 1777, 1811, 1999, 2111, 2221, 3313, 3331, 4111, 4441, 6661, 7177, 7717, 8111, 9199, 10111, 11113

Offset: 1

Reinhard Zumkeller, Feb 05 2012

The terms of A203897 having all divisors in A020449 (in particular, the first 1022 terms) are a subsequence. - M. F. Hasler, May 02 2022

Since 1 and the term itself are divisors, one must only check repdigits and those containing only 1 and another digit. - Michael S. Branicky, May 02 2022

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..500 from M. F. Hasler)

Cf. A027750, A031955, A011531, A106101, A004022, A062634.

Cf. A203897 (an "almost subsequence"), A020449 (primes with only digits 0 & 1), A095048 (number of distinct digits in divisors(n)).

Select[Range[12000],Length[Union[Flatten[IntegerDigits/@Divisors[#]]]]<3&] (* Harvey P. Dale, May 03 2022 *)

select( {is_A206159(n)=#Set(concat([digits(d)|d<-divisors(n)]))<3}, [1..10^4]) \\ M. F. Hasler, May 02 2022

from sympy import divisors
def ok(n):
    digits_used = set()
    for d in divisors(n, generator=True):
        digits_used |= set(str(d))
        if len(digits_used) > 2: return False
    return True
print([k for k in range(1, 9000) if ok(k)]) # Michael S. Branicky, May 02 2022

A095048(a(n)) <= 2.

Terms corrected by Harvey P. Dale, May 02 2022

Edited by N. J. A. Sloane, May 02 2022

cp's OEIS Frontend

A208270 Primes containing a digit 1.

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A293873 Numbers having '13' as substring of their digits.

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A293880 Numbers having '20' as substring of their digits.

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A062634 Numbers k such that every divisor of k contains the digit 1.

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A069715 GCD of digits of n is 1.

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A121042 Smallest divisor of n that is also contained in the decimal representation of n.

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A175688 Numbers k with property that arithmetic mean of its digits is both an integer and one of the digits of k.

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