A037278 -id:A037278 - cp's OEIS frontend

A243360 a(n) = arrange digits of concatenation of divisors of n (A037278, A176558) in decreasing order (in base 10).

1, 21, 31, 421, 51, 6321, 71, 8421, 931, 52110, 111, 6432211, 311, 74211, 55311, 864211, 711, 9863211, 911, 54221100, 73211, 222111, 321, 8644322211, 5521, 632211, 97321, 87442211, 921, 65533211100, 311, 86432211, 333111, 743211, 75531, 986643322111, 731

Offset: 1

Jaroslav Krizek, Jun 04 2014

See A243363 = numbers n such that a(n) = 9876543210.

			For n = 12; divisors of 12: 1, 2, 3, 4, 6, 12; a(12) = 6432211.

Cf. A037278, A176558, A243361, A243362, A243363, A243364.

A243360:=func; [A243360(n): n in [1..100]];

A069872 Numbers k such that k divides the concatenation all divisors in ascending order, i.e., k divides A037278(k).

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 15, 16, 20, 24, 25, 30, 32, 40, 50, 60, 64, 80, 90, 96, 100, 104, 120, 124, 125, 128, 150, 160, 200, 240, 250, 255, 256, 288, 320, 360, 375, 380, 384, 400, 425, 464, 480, 495, 500, 512, 600, 618, 625, 640, 750, 795, 800, 864, 875, 960, 1000

Offset: 1

Amarnath Murthy, Apr 24 2002

All the powers of 2 are terms.

			16 is a term as 16 divides 124816, 24 is a term as 24 divides 1234681224.

Giovanni Resta, Table of n, a(n) for n = 1..1547 (terms <= 10^10, first 500 terms from Harvey P. Dale)

Cf. A037278, A224930.

k:=1; sol:=[];
for u in [1..1000] do D:=Divisors(u); conc:=D[1];
    for u1 in [2..#D] do a:=#Intseq(conc); a1:=#Intseq(D[u1]);conc:=10^a1*conc+D[u1];end for;
     if conc mod u eq 0 then sol[k]:=u; k:=k+1; end if;
end for;
sol; // Marius A. Burtea, Jun 01 2019

Select[Range[1000],Divisible[FromDigits[Flatten[IntegerDigits/@ Divisors[ #]]],#]&] (* Harvey P. Dale, Dec 31 2012 *)

f(n) = my(d=divisors(n), s=""); fordiv(n, d, s = concat(s, Str(d))); eval(s); \\ A037278
isok(n) = f(n) % n == 0; \\ Michel Marcus, Jun 01 2019

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

A190998 Digital root of concatenation of all divisors of n (A037278).

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Jun 15 2011

Also digital root of A034690 (sum of digits of all the divisors of n).

Also digital root of A000203 (sum of divisors of n). - Michel Marcus, Sep 13 2014

			For n = 12: 1 + 2 + 3 + 4 + 6 + 1 + 2 = 19, 1 + 9 = 10, 1 + 0 = 1; a(12) = 1.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A134681, A034690, A190997.

A190998:=proc(n) local d, i, s: d:=numtheory[divisors](n): s:=0: for i from 1 to nops(d) do s:=s+((d[i]-1) mod 9)+1: if(s>=10)then s:=((s-1) mod 9)+1: fi: od: return s: end: seq(A190998(n), n=1..105); # Nathaniel Johnston, Jun 15 2011

A243361 a(n) = arrange digits of concatenation of divisors of n (A037278, A176558) in increasing order in base 10 (zero digits are omitted).

Original entry on oeis.org

1, 12, 13, 124, 15, 1236, 17, 1248, 139, 1125, 111, 1122346, 113, 11247, 11355, 112468, 117, 1123689, 119, 112245, 11237, 111222, 123, 1122234468, 1255, 112236, 12379, 11224478, 129, 111233556, 113, 11223468, 111333, 112347, 13557, 111223346689, 137

Offset: 1

Jaroslav Krizek, Jun 04 2014

See A243362 = sequence of numbers n such that a(n) = 123456789: 54023, 54203, 500407, 23456789… First prime in this sequence is 23456789.

			For n = 20; divisors of 20: 1, 2, 4, 5, 10, 20; a(20) = 112245.

Cf. A037278, A176558, A243360, A243362, A243363, A243364.

A243361:=func; [A243361(n): n in [1..100]];

A248323 Numbers n which appear at least twice in A037278(n), concatenation of their divisors written in base 10.

Original entry on oeis.org

11, 12, 42, 84, 124, 135, 248, 325, 366, 510, 550, 555, 624, 650, 714, 1010, 1111, 1525, 1734, 2510, 3913, 4020, 5100, 5500, 5610, 5625, 8040, 11111, 13515, 16575, 21175, 24104, 25500, 28160, 34170, 35250, 35610, 36800, 37444, 44919, 50100, 51020, 51102, 51250, 52000

Offset: 1

Eric Angelini and M. F. Hasler, Oct 04 2014

Overlapping is allowed, so a(1) = 11 is in the sequence, with concatenated divisors A037278(11) = "111".
All repunits (10^k-1)/9 = A000042(k) = A002275(k) with even k = number of digits (as to be divisible by 11) but not multiples of 3, i.e., k in A047235, have divisors {1, 11, ..., 1010...101, 1111...111} and therefore are in this sequence.
Numbers n = floor(10^(8+3k)/7), k>=0, also belong to this sequence; for k>=2m, the number n appears (at least) m+2 times in A037278(n). [Found by extending results from Hans Havermann.]
The smallest terms that appear more than twice in the concatenation are 1111, 400200, 800400, 28571428, all 3 times, and 42857142, 4 times. - Hans Havermann, Oct 05 2014

			The divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and "42" appears twice in their concatenation A037278(42) = "12367142142".

Chai Wah Wu, Table of n, a(n) for n = 1..340
E. Angelini, Divisors showing a string, SeqFan list, Oct 03 2014

Cf. A037278.

is(n)={d=concat(apply(digits,divisors(n)));n=digits(n);for(j=0,#d-#n-1,for(i=1,#n,d[i+j]==n[i]||next(2));return(1))}

from sympy import divisors
import re
A248323_list = [n for n in range(1,10**7) if len(list(re.finditer('(?='+str(n)+')',''.join([str(d) for d in divisors(n)])))) > 1]
# Chai Wah Wu, Nov 01 2014

A357692 Integers k such that A037278(k) is a term of A175252.

Original entry on oeis.org

1, 2, 4, 15, 16, 25, 60, 90, 100, 124, 150, 240, 375, 384, 600, 618, 625, 960, 1536, 3330, 3750, 4650, 5760, 10000, 10500, 10752, 15000, 16384, 17500, 24576, 25600, 40000, 49500, 62500, 102400, 139200, 168750, 198400, 323280, 526848, 960000, 1179648, 1248000, 1369125

Offset: 1

Michel Marcus, Oct 10 2022

			A037278(4) = 124, a term of A175252.
A037278(15) = 13515, a term of A175252.
A037278(16) = 124816, a term of A175252.
A037278(25) = 1525, a term of A175252.

David A. Corneth, Table of n, a(n) for n = 1..81

Subsequence of A069872.

Cf. A037278, A175252.

is(n, {u = 10^5}) = {my(e = eval(concat(concat([""], divisors(n))))); if(e % n != 0, return(0); ); my(oldu = u, s, d); u = min(e, u); s = ""; d = divisors(factor(e, u)); d = select(x -> x <= u, d); for(i = 1, #d, s=concat(s, Str(d[i])); if(eval(s) == e, return(1)); if(eval(s) > e, return(0)); ); is(n, 10*oldu); } \\ David A. Corneth, Oct 10 2022, adapted from Michel Marcus' isok at A175252

More terms from David A. Corneth, Oct 10 2022

A231817 Multiplicative digital root of concatenation of all divisors of n (A037278).

Original entry on oeis.org

1, 2, 3, 8, 5, 8, 7, 8, 4, 0, 1, 6, 3, 0, 5, 0, 7, 0, 9, 0, 8, 8, 6, 8, 0, 4, 6, 0, 8, 0, 3, 0, 4, 6, 0, 0, 2, 8, 8, 0, 4, 0, 2, 0, 0, 6, 6, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 6, 4, 0, 0, 0, 0, 8, 0, 2, 0, 7, 0, 2, 8, 0, 0, 8, 0, 8, 0, 0, 6, 8, 0, 0, 0, 0

Offset: 1

Jaroslav Krizek, Nov 13 2013

Also multiplicative digital root of A190997 (product of digits of all the divisors of n) or A007955 (product of divisors of n).
Conjecture: a(n) = 0 for almost all n.
793 of the first 1000 terms are zeros, and 9147 out of the first 10000 terms are zeros. - Harvey P. Dale, Jul 30 2019

			For n=12: 1*2*3*4*6*1*2=288, 2*8*8=128, 1*2*8=16, 1*6=6; a(12)=6.

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

Cf. A037278, A007955, A190998 (associative digital root of digits of all the divisors of n), A031347 (multiplicative digital root of n).

Table[NestWhile[Times@@IntegerDigits[#]&,Times@@Flatten[ IntegerDigits/@ Divisors[ n]], #>9&],{n,90}] (* Harvey P. Dale, Jul 30 2019 *)

A027750 Triangle read by rows in which row n lists the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 3, 6, 9, 18, 1, 19, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 29

Offset: 1

N. J. A. Sloane

Or, in the list of natural numbers (A000027), replace n with its divisors.
This gives the first elements of the ordered pairs (a,b) a >= 1, b >= 1 ordered by their product ab.
Also, row n lists the largest parts of the partitions of n whose parts are not distinct. - Omar E. Pol, Sep 17 2008
Concatenation of n-th row gives A037278(n). - Reinhard Zumkeller, Aug 07 2011
{A210208(n,k): k=1..A073093(n)} subset of {T(n,k): k=1..A000005(n)} for all n. - Reinhard Zumkeller, Mar 18 2012
Row sums give A000203. Right border gives A000027. - Omar E. Pol, Jul 29 2012
Indices of records are in A006218. - Irina Gerasimova, Feb 27 2013
The number of primes in the n-th row is omega(n) = A001221(n). - Michel Marcus, Oct 21 2015
The row polynomials P(n,x) = Sum_{k=1..A000005(n)} T(n,k)*x^k with composite n which are irreducible over the integers are given in A292226. - Wolfdieter Lang, Nov 09 2017
T(n,k) is also the number of parts in the k-th partition of n into equal parts (see example). - Omar E. Pol, Nov 20 2019
Let there be an infinite number of tiles, each labeled with a positive integer m, initially placed on square m of an infinite 1D board. At step n, the leftmost unblocked tile (i.e., the top tile of the leftmost nonempty stack) moves forward exactly m squares, where m is its label. Tiles that land on the same square form a stack, and only the top tile of any stack may move. This sequence records the label m of the tile that moves at step n. - Ali Sada, May 23 2025
All divisors of a positive integer n form a finite set. Extending divisibility to n = 0 by using the definition (k|n <=> exists m such that m*k = n) makes the set of divisors infinite, suggesting the definition was not intended for zero, as arithmetic functions typically apply to n >= 1. So to preserve a core property when generalizing (cardinality), one can define divisors of n >= 0 as the fixed points of the greatest common divisor on the set [n] = {0, 1, 2, ..., n}. By this definition, the divisors of 0 are {0}, since 0|0 and gcd(0, 0) = 0. This definition is not circular because the gcd can be effectively calculated using the Euclidean algorithm. (Cf. links.) - Peter Luschny, Jun 02 2025

			Triangle begins:
  1;
  1, 2;
  1, 3;
  1, 2, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 4, 8;
  1, 3, 9;
  1, 2, 5, 10;
  1, 11;
  1, 2, 3, 4, 6, 12;
  ...
For n = 6 the partitions of 6 into equal parts are [6], [3,3], [2,2,2], [1,1,1,1,1,1], so the number of parts are [1, 2, 3, 6] respectively, the same as the divisors of 6. - _Omar E. Pol_, Nov 20 2019

Franklin T. Adams-Watters, Rows 1..1000, flattened
Franklin T. Adams-Watters, Rows 1..10000
Peter Luschny, A Python program using gcd for n >= 0.
Omar E. Pol, Illustration of initial terms, (2009).
Eric Weisstein's World of Mathematics, Divisor
Wikipedia, Table of divisors
Index entries for sequences related to divisors of numbers

Cf. A000005 (row length), A001221, A027749, A027751, A056534, A056538, A127093, A135010, A161700, A163280, A240698 (partial sums of rows), A240694 (partial products of rows), A247795 (parities), A292226, A244051.

a027750 n k = a027750_row n !! (k-1)
a027750_row n = filter ((== 0) . (mod n)) [1..n]
a027750_tabf = map a027750_row [1..]
-- Reinhard Zumkeller, Jan 15 2011, Oct 21 2010

Magma
```
[Divisors(n) : n in [1..20]];
```

seq(op(numtheory:-divisors(a)), a = 1 .. 20) # Matt C. Anderson, May 15 2017

Flatten[ Table[ Flatten [ Divisors[ n ] ], {n, 1, 30} ] ]

v=List();for(n=1,20,fordiv(n,d,listput(v,d)));Vec(v) \\ Charles R Greathouse IV, Apr 28 2011

from sympy import divisors
for n in range(1, 16):
    print(divisors(n)) # Indranil Ghosh, Mar 30 2017

a(A006218(n-1) + k) = k-divisor of n, 1 <= k <= A000005(n). - Reinhard Zumkeller, May 10 2006

T(n,k) = n / A056538(n,k) = A056538(n,n-k+1), 1 <= k <= A000005(n). - Reinhard Zumkeller, Sep 28 2014

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A095050 Numbers such that all ten digits are needed to write all positive divisors in decimal representation.

Original entry on oeis.org

108, 216, 270, 304, 306, 312, 324, 360, 380, 406, 432, 450, 504, 540, 570, 608, 612, 624, 630, 648, 654, 702, 708, 714, 720, 728, 756, 760, 780, 810, 812, 864, 870, 900, 910, 912, 918, 924, 936, 945, 954, 972, 980, 1008, 1014, 1026, 1032, 1036, 1038

Offset: 1

Reinhard Zumkeller, May 28 2004

A095048(a(n)) = 10.
Numbers n such that A037278(n), A176558(n) and A243360(n) contain 10 distinct digits. - Jaroslav Krizek, Jun 19 2014
Once a number is in the sequence, then all its multiples will be there too. The list of primitive terms begin: 108, 270, 304, 306, 312, 360, 380, ... - Michel Marcus, Jun 20 2014
Pandigital numbers A050278 and A171102 are subsequences. - Michel Marcus, May 01 2020

			Divisors of 108 are: [1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108] where all digits can be found.

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

Cf. A095048, A059436 (subsequence), A206159.
Cf. A243543 (the smallest number m whose list of divisors contains n distinct digits).
Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: 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. - Jaroslav Krizek, Jun 19 2014
Cf. A050278, A171102.

import Data.List (elemIndices)
a095050 n = a095050_list !! (n-1)
a095050_list = map (+ 1) $ elemIndices 10 $ map a095048 [1..]
-- Reinhard Zumkeller, Feb 05 2012

q:= n-> is({$0..9}=map(x-> convert(x, base, 10)[], numtheory[divisors](n))):
select(q, [$1..2000])[];  # Alois P. Heinz, Oct 28 2021

Select[Range@2000, 1+Union@@IntegerDigits@Divisors@# == Range@10 &] (* Hans Rudolf Widmer, Oct 28 2021 *)

isok(m)=my(d=divisors(m), v=[1]); for (k=2, #d, v = Set(concat(v, digits(d[k]))); if (#v == 10, return (1));); #v == 10; \\ Michel Marcus, May 01 2020

from sympy import divisors
def ok(n):
    digits_used = set()
    for d in divisors(n):
        digits_used |= set(str(d))
    return len(digits_used) == 10
print([k for k in range(1040) if ok(k)]) # Michael S. Branicky, Oct 28 2021

a(n) ~ n. - Charles R Greathouse IV, Nov 16 2022

A243534 Numbers n such that the list of all divisors of n contains only 1 distinct digit (in base 10).

Original entry on oeis.org

1, 11, 1111111111111111111, 11111111111111111111111

Offset: 1

Jaroslav Krizek, Jun 13 2014

Union of 1 and A004022 (prime repunits).
The next term has 317 digits.
Numbers n such that A037278(n), A176558(n) and A243360(n) contain only 1 distinct digit.

			11 is in sequence because the list of the divisors of 11: (1, 11) contains only 1 distinct digit.

Cf. A095048, A037278, A176558, A243360.
Sequences of numbers n such that the list of divisors of n contains k distinct digits for 1 <= k <= 10: 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. A243543.

[Row n = 1 …10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=1;A(n)); Arrangement of column B]

a(1) = 1; for n >= 2, a(n+1) = A004022 (prime repunits).

cp's OEIS Frontend

A243360 a(n) = arrange digits of concatenation of divisors of n (A037278, A176558) in decreasing order (in base 10).

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Magma

A069872 Numbers k such that k divides the concatenation all divisors in ascending order, i.e., k divides A037278(k).

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Magma

Mathematica

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A190998 Digital root of concatenation of all divisors of n (A037278).

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Maple

A243361 a(n) = arrange digits of concatenation of divisors of n (A037278, A176558) in increasing order in base 10 (zero digits are omitted).

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Magma

A248323 Numbers n which appear at least twice in A037278(n), concatenation of their divisors written in base 10.

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PARI

Python

A357692 Integers k such that A037278(k) is a term of A175252.

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PARI

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A231817 Multiplicative digital root of concatenation of all divisors of n (A037278).

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Mathematica

A027750 Triangle read by rows in which row n lists the divisors of n.

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