A243360 -id:A243360 - cp's OEIS frontend

A243543 Smallest number whose list of divisors contains n distinct digits (in base 10).

1, 2, 4, 6, 12, 18, 36, 72, 54, 108

Offset: 1

Jaroslav Krizek, Jun 19 2014

Finite sequence with 10 terms.

			a(9) = 54 because 54 is the smallest number whose list of divisors contains 9 distinct digits; the list of divisors of 54: (1, 2, 3, 6, 9, 18, 27, 54) contains 9 distinct digits (1, 2, 3, 4, 5, 6, 7, 8, 9).

Cf. Sequences of numbers n such that list of divisors of n contains k distinct digits: k = 1: A243534; k = 2: A243535; k = 3: A243536; k = 4: A243537; k = 5: A243538; k = 6: A243539; k = 7: A243540; k = 8: A243541; k = 9: A243542; k = 10: A095050.

Cf. also A095048, A037278, A176558, A243360.

A257219 Numbers that have at least one divisor containing the digit 2 in base 10.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 87, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 106, 108

Offset: 1

Jaroslav Krizek, Apr 20 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 2.
Sequences of numbers k whose concatenation of divisors contains a digit j in base 10 for 0 <= j <= 9: A209932 for j = 0, A000027 for j = 1, A257219 for j = 2, A257220 for j = 3, A257221 for j = 4, A257222 for j = 5, A257223 for j = 6, A257224 for j = 7, A257225 for j = 8, A257226 for j = 9.
All even numbers and all numbers which have a digit "2" themselves are trivially in this sequence. The first terms not of this form are the odd multiples of odd numbers between 21 and 29: { 63 = 3*21, 69 = 3*23, 75 = 3*25, 81 = 3*27, 87 = 3*29, 105 = 5*21, 115 = 5*23, 135 = 5*27, 145 = 5*29, ...}. - M. F. Hasler, Apr 22 2015
A011532 (numbers that contain a 2) is a subsequence. - Michel Marcus, May 19 2015

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 2.
In the same way all even numbers have the divisor 2 and thus are in this sequence; numbers N in { 20,...,29, 120,...,129, 200,...,299 } have the digit 2 in N which is divisor of itself. - _M. F. Hasler_, Apr 22 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037278, A176558, A243360, A256824.

[n: n in [1..1000] | [2] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))]

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 2] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)

is(n)=!bittest(n,0)||setsearch(Set(digits(n)),2)||fordiv(n,d,setsearch(Set(digits(d)),2)&&return(1)) \\ M. F. Hasler, Apr 22 2015

a(n) ~ n. - Charles R Greathouse IV, Apr 22 2015

A257220 Numbers that have at least one divisor containing the digit 3 in base 10.

Original entry on oeis.org

3, 6, 9, 12, 13, 15, 18, 21, 23, 24, 26, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 46, 48, 51, 52, 53, 54, 57, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 78, 81, 83, 84, 86, 87, 90, 91, 92, 93, 96, 99, 102, 103, 104, 105, 106, 108

Offset: 1

Jaroslav Krizek, Apr 20 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 3.

Sequences of numbers k whose concatenation of divisors contains a digit j in base 10 for 0 <= j <= 9: A209932 for j = 0, A000027 for j = 1, A257219 for j = 2, A257220 for j = 3, A257221 for j = 4, A257222 for j = 5, A257223 for j = 6, A257224 for j = 7, A257225 for j = 8, A257226 for j = 9.

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037278, A176558, A243360, A256824.

[n: n in [1..1000] | [3] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 3] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)

is(n)=fordiv(n,d, if(setsearch(Set(digits(d)),3), return(1))); 0 \\ Charles R Greathouse IV, Apr 30 2015

a(n) ~ n. - Charles R Greathouse IV, Apr 30 2015

A257221 Numbers that have at least one divisor containing the digit 4 in base 10.

Original entry on oeis.org

4, 8, 12, 14, 16, 20, 24, 28, 32, 34, 36, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 52, 54, 56, 60, 64, 68, 70, 72, 74, 76, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 108, 112, 114, 116, 120, 123, 124, 126, 128, 129, 132, 134, 135, 136, 138, 140, 141

Offset: 1

Jaroslav Krizek, Apr 20 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 4.

Sequences of numbers k whose concatenation of divisors contains a digit j in base 10 for 0 <= j <= 9: A209932 for j = 0, A000027 for j = 1, A257219 for j = 2, A257220 for j = 3, A257221 for j = 4, A257222 for j = 5, A257223 for j = 6, A257224 for j = 7, A257225 for j = 8, A257226 for j = 9.

			16 is in sequence because the list of divisors of 16: (1, 2, 4, 8, 16) contains digit 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A037278, A176558, A243360, A256824.

[n: n in [1..1000] | [4] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))]

Select[Range@ 141, Part[Plus @@ DigitCount@ Divisors@ #, 4] > 0 &] (* Michael De Vlieger, Apr 20 2015 *)
Select[Range[200],Count[Flatten[IntegerDigits/@Divisors[#]],4]>0&] (* Harvey P. Dale, May 05 2022 *)

is(n)=fordiv(n,d, if(setsearch(Set(digits(d)),4), return(1))); 0 \\ Charles R Greathouse IV, Apr 30 2015

a(n) ~ n. - Charles R Greathouse IV, Apr 30 2015

A257222 Numbers that have at least one divisor containing the digit 5 in base 10.

Original entry on oeis.org

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 65, 70, 75, 80, 85, 90, 95, 100, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 118, 120, 125, 130, 135, 140, 145, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 162, 165

Offset: 1

Jaroslav Krizek, May 05 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 5.

Sequences of numbers k whose concatenation of divisors contains a digit j in base 10 for 0 <= j <= 9: A209932 for j = 0, A000027 for j = 1, A257219 for j = 2, A257220 for j = 3, A257221 for j = 4, A257222 for j = 5, A257223 for j = 6, A257224 for j = 7, A257225 for j = 8, A257226 for j = 9.

			20 is in sequence because the list of divisors of 20: (1, 2, 4, 5, 10, 20) contains digit 5.

Cf. A037278, A176558, A243360, A256824.

[n: n in [1..1000] | [5] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 5] > 0 &]
Select[Range[200],Max[DigitCount[Divisors[#],10,5]]>0&] (* Harvey P. Dale, Sep 15 2018 *)

is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 5), return(1))); 0

use ntheory ":all"; for my $n (1..1000) { say $n if scalar(grep {/5/} divisors($n)) } # Dana Jacobsen, May 07 2015

use ntheory ":all"; my @a257222 = grep { scalar(grep {/5/} divisors($)) } 1..1000; # _Dana Jacobsen, May 07 2015

from sympy import divisors
A257222_list = [n for n in range(1,10**3) if '5' in set().union(*(set(str(d)) for d in divisors(n,generator=True)))] # Chai Wah Wu, May 06 2015

a(n) ~ n.

Mathematica and PARI programs with assistance from Michael De Vlieger and Charles R Greathouse IV, respectively.

A257223 Numbers that have at least one divisor containing the digit 6 in base 10.

Original entry on oeis.org

6, 12, 16, 18, 24, 26, 30, 32, 36, 42, 46, 48, 52, 54, 56, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 72, 76, 78, 80, 84, 86, 90, 92, 96, 102, 104, 106, 108, 112, 114, 116, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 144, 146, 150, 152, 156, 160, 161, 162

Offset: 1

Jaroslav Krizek, May 05 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 6.

A011536 (numbers that contain a 6) is a subsequence. - Michel Marcus, May 25 2015

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 6.

Cf. A037278, A176558, A243360, A256824.

Cf. similar sequences with another digit: A209932 (0), A000027 (1), A257219 (2), A257220 (3), A257221 (4), A257222 (5), A257224 (7), A257225 (8), A257226 (9).

[n: n in [1..1000] | [6] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 6] > 0 &]
Select[Range[200],Count[Flatten[IntegerDigits/@Divisors[#]],6]>0&] (* Harvey P. Dale, Nov 05 2021 *)

is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 6), return(1))); 0

a(n) ~ n.

Mathematica and PARI programs with assistance from Michael De Vlieger and Charles R Greathouse IV, respectively.

A257224 Numbers that have at least one divisor containing the digit 7 in base 10.

Original entry on oeis.org

7, 14, 17, 21, 27, 28, 34, 35, 37, 42, 47, 49, 51, 54, 56, 57, 63, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 84, 85, 87, 91, 94, 97, 98, 102, 105, 107, 108, 111, 112, 114, 117, 119, 126, 127, 133, 134, 135, 136, 137, 140, 141, 142, 144, 146, 147, 148

Offset: 1

Jaroslav Krizek, May 05 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 7.

A011537 (numbers that contain a 7) is a subsequence. - Michel Marcus, May 25 2015

			14 is in sequence because the list of divisors of 14: (1, 2, 7, 14) contains digit 7.

Cf. A037278, A176558, A243360, A256824.

Cf. similar sequences with another digit: A209932 (0), A000027 (1), A257219 (2), A257220 (3), A257221 (4), A257222 (5), A257223 (6), A257225 (8), A257226 (9).

[n: n in [1..1000] | [7] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 7] > 0 &]

is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 7), return(1))); 0

from itertools import count, islice
from sympy import divisors
def A257224_gen(): return filter(lambda n:any('7' in str(d) for d in divisors(n, generator=True)), count(1))
A257224_list = list(islice(A257224_gen(), 60)) # Chai Wah Wu, Dec 27 2021

a(n) ~ n.

Mathematica and PARI programs with assistance from Michael De Vlieger and Charles R Greathouse IV, respectively.

A257225 Numbers that have at least one divisor containing the digit 8 in base 10.

Original entry on oeis.org

8, 16, 18, 24, 28, 32, 36, 38, 40, 48, 54, 56, 58, 64, 68, 72, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 96, 98, 104, 108, 112, 114, 116, 118, 120, 126, 128, 136, 138, 140, 144, 148, 152, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178

Offset: 1

Jaroslav Krizek, May 07 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 8.

A011538 (numbers that contain an 8) is a subsequence. - Michel Marcus, May 19 2015

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 8.

Cf. A037278, A176558, A243360, A256824.

Cf. similar sequences with another digit: A209932 (0), A000027 (1), A257219 (2), A257220 (3), A257221 (4), A257222 (5), A257223 (6), A257224 (7), A257226 (9).

[n: n in [1..1000] | [8] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

select(t -> has(map(convert,numtheory:-divisors(t),base,10),8), [$1..200]); # Robert Israel, May 14 2015

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 8] > 0 &]
Select[Range[200],SequenceCount[Flatten[IntegerDigits/@Divisors[#]],{8}]> 0&] (* Harvey P. Dale, Aug 02 2021 *)

is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 8), return(1))); 0

from itertools import count, islice
from sympy import divisors
def A257225_gen(): return filter(lambda n:any('8' in str(d) for d in divisors(n, generator=True)), count(1))
A257225_list = list(islice(A257225_gen(), 58)) # Chai Wah Wu, Dec 27 2021

a(n) ~ n.

Mathematica and PARI programs with assistance from Michael De Vlieger and Charles R Greathouse IV, respectively.

A257226 Numbers that have at least one divisor containing the digit 9 in base 10.

Original entry on oeis.org

9, 18, 19, 27, 29, 36, 38, 39, 45, 49, 54, 57, 58, 59, 63, 69, 72, 76, 78, 79, 81, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 108, 109, 114, 116, 117, 118, 119, 126, 129, 133, 135, 138, 139, 144, 145, 147, 149, 152, 153, 156, 158, 159, 162, 169, 171, 174

Offset: 1

Jaroslav Krizek, May 29 2015

Numbers k whose concatenation of divisors A037278(k), A176558(k), A243360(k) or A256824(k) contains a digit 9.

A011539 (numbers that contain a 9) is a subsequence.

			18 is in sequence because the list of divisors of 18: (1, 2, 3, 6, 9, 18) contains digit 9.

Cf. A037278, A176558, A243360, A256824.

Cf. similar sequences with another digit: A209932 (0), A000027 (1), A257219 (2), A257220 (3), A257221 (4), A257222 (5), A257223 (6), A257224 (7), A257225 (8).

[n: n in [1..1000] | [9] subset Setseq(Set(Sort(&cat[Intseq(d): d in Divisors(n)])))];

Select[Range@108, Part[Plus @@ DigitCount@ Divisors@ #, 9] > 0 &] (* after Michael De Vlieger *)

is(n)=fordiv(n, d, if(setsearch(Set(digits(d)), 9), return(1))); 0 \\ after Charles R Greathouse IV

from itertools import count, islice
from sympy import divisors
def A257226_gen(): return filter(lambda n:any('9' in str(d) for d in divisors(n,generator=True)),count(1))
A257226_list = list(islice(A257226_gen(),20)) # Chai Wah Wu, Dec 27 2021

a(n) ~ n.

A243363 Numbers with divisors containing all the digits 0-9 and each digit appears exactly once (in base 10).

Original entry on oeis.org

203457869, 203465789, 203465897, 203468579, 203475869, 203478659, 203485697, 203485769, 203495867, 203548967, 203564897, 203568947, 203574689, 203584679, 203584769, 203594687, 203596847, 203598467, 203645879, 203645987, 203648957, 203654987, 203659487, 203674589

Offset: 1

Jaroslav Krizek, Jun 04 2014

Primes made up of distinct digits except 1.
There are no composite numbers with divisors containing all the digits 0-9 and each digit appears exactly once.
Subsequence of A029743 (primes with distinct digits).
Numbers n such that A243360(n) = 9876543210.
Sequence contains 19558 terms, the last term is a(19558) = 987625403.

Michael S. Branicky, Table of n, a(n) for n = 1..19558 (full sequence)

Cf. A037278, A176558, A243360, A243361, A243362, A243364.

[n: n in [1..203457879] | Seqint(Sort(&cat[(Intseq(k)): k in Divisors(n)])) eq 9876543210];

Select[Range[203*10^6,204*10^6],Sort[Flatten[IntegerDigits/@ Divisors[#]]] == Range[0,9]&] (* Harvey P. Dale, Aug 22 2016 *)

# generates entire sequence
from sympy import isprime
from itertools import permutations as perms
dist = (int("".join(p)) for p in perms("023456789", 9) if p[0] != "0")
afull = [k for k in dist if isprime(k)]
print(afull[:24]) # Michael S. Branicky, Aug 04 2022

cp's OEIS Frontend

A243543 Smallest number whose list of divisors contains n distinct digits (in base 10).

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A257219 Numbers that have at least one divisor containing the digit 2 in base 10.

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A257220 Numbers that have at least one divisor containing the digit 3 in base 10.

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A257221 Numbers that have at least one divisor containing the digit 4 in base 10.

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A257222 Numbers that have at least one divisor containing the digit 5 in base 10.

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A257223 Numbers that have at least one divisor containing the digit 6 in base 10.

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A257224 Numbers that have at least one divisor containing the digit 7 in base 10.

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