A037278 -id:A037278 - cp's OEIS frontend

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

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

A037283 Replace n with concatenation of its odd divisors.

Original entry on oeis.org

1, 1, 13, 1, 15, 13, 17, 1, 139, 15, 111, 13, 113, 17, 13515, 1, 117, 139, 119, 15, 13721, 111, 123, 13, 1525, 113, 13927, 17, 129, 13515, 131, 1, 131133, 117, 15735, 139, 137, 119, 131339, 15, 141, 13721, 143, 111, 13591545, 123, 147, 13, 1749, 1525, 131751

Offset: 1

N. J. A. Sloane

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

Cf. A182469, A037278, A037284, A037285.

a037283 = read . concat . (map show) . a182469_row :: Integer -> Integer
-- Reinhard Zumkeller, May 01 2012

dtn[ L_ ] := Fold[ 10#1+#2&, 0, L ] Array[ dtn[ Flatten[ Map[ IntegerDigits, Select[ Divisors[ # ], OddQ ] ] ] ]&, 50 ]
cod[n_]:=FromDigits[Flatten[IntegerDigits/@Select[Divisors[n],OddQ]]]; Array[cod,60] (* Harvey P. Dale, Jan 24 2014 *)

from sympy import divisors
def a(n): return int("".join(str(d) for d in divisors(n) if d%2==1))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Dec 31 2020

More terms from Erich Friedman

A134681 Number of digits of all the divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 3, 7, 3, 5, 5, 6, 3, 7, 3, 8, 5, 6, 3, 10, 4, 6, 5, 8, 3, 11, 3, 8, 6, 6, 5, 12, 3, 6, 6, 11, 3, 11, 3, 9, 8, 6, 3, 14, 4, 9, 6, 9, 3, 11, 6, 11, 6, 6, 3, 18, 3, 6, 8, 10, 6, 12, 3, 9, 6, 12, 3, 17, 3, 6, 9, 9, 6, 12, 3, 15, 7, 6, 3, 18, 6, 6, 6, 12, 3, 18, 6, 9, 6, 6, 6, 18

Offset: 1

Reinhard Zumkeller, Nov 06 2007

Also number of digits of the concatenation of all divisors of n (A037278). - Jaroslav Krizek, Jun 15 2011

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

Cf. A000005, A037278, A055642, A190997, A034690, A134682.

A134681 := proc(n)
    add(A055642(d),d=numtheory[divisors](n)) ;
end proc:
seq(A134681(n),n=1..80) ; # R. J. Mathar, Feb 21 2025

Array[Total[IntegerLength[Divisors[#]]]&,100] (* Harvey P. Dale, Jun 08 2013 *)

a(n) = sumdiv(n, d, #digits(d)); \\ Michel Marcus, Sep 01 2023

from sympy import divisors
def a(n): return sum(len(str(d)) for d in divisors(n, generator=True))
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Nov 03 2023

a(n) = A055642(A037278(n)) = Number of digits of the concatenation of all divisors of n.
From Sida Li, Sep 01 2023: (Start)
a(n) = Sum_{d divides n} (floor(log_10(d))+1).
log_10(Product_{d divides n} d) <= a(n) <= log_10(Product_{d divides n} d) + sigma_0(n), where sigma_0(n) = A000005(n).
Equivalently, sigma_0(n)*log_10(n)/2 <= a(n) <= sigma_0(n)*log_10(n)/2 + sigma_0(n), obtained by formula in A007955.
For x >= 5, c2*log(x)^2 + c1*log(x) + c0 <= (1/x)*Sum_{n<=x} a(n) <= c2*log(x)^2 + (c1+1)*log(x) + 2*c0, where c2 = 1/(2*log(10)), c1 = (gamma-1)/log(10), c0 = 2*gamma-1, and gamma is Euler's constant. This is obtained by hyperbola trick for Sum_{n<=x} sigma_0(n), and Abel partial summation on Sum_{n<=x} sigma_0(n)*log(n). (End)

New name from Jaroslav Krizek, Jun 15 2011

A176554 Numbers n such that concatenations of divisors of n are nonprime.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81

Offset: 1

Jaroslav Krizek, Apr 20 2010

See A037278(n) = concatenation of divisors of n. See A176556 for corresponding values of concatenations. Complement of A176553(n) for n >= 2.

			a(6) = 8: divisors of 8 are 1,2,4,8 and their concatenation 1248 is nonprime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Select[Range[100],!PrimeQ[FromDigits[Flatten[IntegerDigits/@ Divisors[ #]]]]&] (* Harvey P. Dale, Jul 09 2021 *)

Edited and extended by Charles R Greathouse IV, Apr 30 2010

A209931 Numbers k such that smallest digit of all divisors of k is 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 111

Offset: 1

Jaroslav Krizek, Mar 20 2012

Also numbers k such that smallest digit of concatenation of all divisors of k (A037278 or A176558) is 1.
Sequence is not the same as A052382, first deviation is at a(173): A052382(173) = 212, a(173) = 213. [Corrected by Michael S. Branicky, Jul 01 2025.]
Sequence is not the same as A067251, first deviation is at a(91): A067251 (91) = 101, a(91) = 111.
Complement of A209932.

			Number 24 is in sequence because smallest digit of all divisors of 24 (1, 2, 4, 8, 3, 6, 12, 24) is 1.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A052382, A067251, A209929 (smallest digit of all divisors of n).

isA209931 := proc(n)
    digsdiv := {} ;
    for d in numtheory[divisors](n) do
        dgs := convert(convert(d,base,10),set) ;
        digsdiv := digsdiv union dgs ;
    end do:
    if 0 in digsdiv then
        false;
    else
        true ;
    end if;
end proc:
A209931 := proc(n)
    option remember;
    if n =1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA209931(a) then
                return a;
            end if;
        end do;
    end if;
end proc:
seq(A209931(n),n=1..120) ;# R. J. Mathar, Dec 28 2023

Select[Range[100], Min[IntegerDigits[Divisors[#]]] == 1 &] (* Paolo Xausa, Jul 03 2025 *)

from sympy import divisors
def ok(n): return all('0' not in str(d) for d in divisors(n, generator=True))
print([k for k in range(1, 112) if ok(k)]) # Michael S. Branicky, Jul 01 2025

A037277 Replace n with concatenation of its divisors >1.

Original entry on oeis.org

0, 2, 3, 24, 5, 236, 7, 248, 39, 2510, 11, 234612, 13, 2714, 3515, 24816, 17, 236918, 19, 2451020, 3721, 21122, 23, 234681224, 525, 21326, 3927, 2471428, 29, 2356101530, 31, 2481632, 31133, 21734, 5735, 23469121836, 37, 21938, 31339

Offset: 1

N. J. A. Sloane

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

Cf. A027750, A037284, A037278, A106708.

a037277 1 = 0
a037277 n = read $ concat $ map show $ tail $ a027750_row n
-- Reinhard Zumkeller, May 01 2012, Feb 07 2011

FromDigits[Flatten[IntegerDigits/@Rest[Divisors[#]]]]&/@Range[40] (* Harvey P. Dale, Nov 06 2011 *)

from sympy import divisors
def a(n):
  divisors_gt1 = divisors(n)[1:]
  if len(divisors_gt1) == 0: return 0
  else: return int("".join(str(d) for d in divisors_gt1))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Dec 31 2020

More terms from Erich Friedman

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Python

A037283 Replace n with concatenation of its odd divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Python

Extensions

A134681 Number of digits of all the divisors of n.

Views