A256824 -id:A256824 - cp's OEIS frontend

A256826 a(n) = the smallest number k such that A256824(k) = A256825(n).

1, 101, 2, 3, 41, 5, 61, 7, 181, 19, 202, 103, 23, 401, 4, 43, 505, 25, 15, 451, 601, 122, 163, 461, 1661, 107, 127, 37, 47, 157, 67, 1801, 281, 83, 1481, 5581, 1861, 187, 109, 29, 9, 149, 59, 619, 79, 89, 2003, 404, 403, 123, 10, 503, 115, 4051, 12451, 453

Offset: 1

Jaroslav Krizek, Apr 13 2015

A256824(n) = reverse concatenation of distinct digits of all divisors of n in base 10, A256825(n) = possible values of A256824(m) in increasing order.

Finite sequence with 512 terms. Maximal term is a(185) = 88511.

			a(11) = 202 because 202 is the smallest number k such that reverse concatenation of distinct digits of all divisors of k (i.e. 1, 2, 101, 202) in base 10 = A256824(k) = A256824(202) = A256825(11) = 210.

Jaroslav Krizek, Table of n, a(n) for n = 1..512 (complete list)

Cf. A009995, A095050, A243534, A256824, A256825.

A256826:=func; [A256826(n): n in[A256825(n)]];

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(), 20)) # 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(), 20)) # 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.

A256825 Numbers with digits in strictly decreasing order containing digit 1.

Original entry on oeis.org

1, 10, 21, 31, 41, 51, 61, 71, 81, 91, 210, 310, 321, 410, 421, 431, 510, 521, 531, 541, 610, 621, 631, 641, 651, 710, 721, 731, 741, 751, 761, 810, 821, 831, 841, 851, 861, 871, 910, 921, 931, 941, 951, 961, 971, 981, 3210, 4210, 4310, 4321, 5210, 5310, 5321

Offset: 1

Jaroslav Krizek, Apr 10 2015

a(n) = possible values of A256824(m) in increasing order where A256824(m) = reverse concatenation of distinct digits of all divisors of m in base 10.
There are precisely 512 terms. Maximal term is 9876543210.
Subsequence of A009995 (numbers with digits in strictly decreasing order).
See A256826 - the smallest number k such that A256824(k) = a(n).

			21 is in sequence because there are numbers m such that A256824(m) = 21 (for m = 2, 22, 121, 211, 2111, ...).

Jaroslav Krizek, Table of n, a(n) for n = 1..512 (complete list)

Cf. A009995, A095050, A243534, A256824, A256826.

[Row n = 1 … 1023; Column A: A(n) = A009995(n); Column B: B(n) =  =IF(FIND("1";A(n);1)>0;A(n)); Arrangement of column B]

lista(nn) = for (n=1, nn, if ((d = digits(n)) && ((sd=vecsort(d,,8))==Vecrev(d)) && vecsearch(sd, 1), print1(n, ", "))); \\ Michel Marcus, Apr 11 2015

cp's OEIS Frontend

A256826 a(n) = the smallest number k such that A256824(k) = A256825(n).

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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.