A037278 -id:A037278 - cp's OEIS frontend

A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, otherwise a(n) = n.

1, 2, 3, 2, 5, 23, 7, 24, 3, 25, 11, 2346, 13, 27, 35, 248, 17, 2369, 19, 24510, 37, 211, 23, 2346812, 5, 213, 39, 24714, 29, 23561015, 31, 24816, 311, 217, 57, 234691218, 37, 219, 313, 24581020, 41, 23671421, 43, 241122, 35915, 223, 47, 23468121624, 7

Offset: 1

N. J. A. Sloane

			Divisors of 12 are 1,2,3,4,6,12, so a(12)=2346.

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

Cf. A037278, A120712, A106708.

A037279 := proc(n) local dvs ; if isprime(n) or n = 1 then n; else dvs := [op(numtheory[divisors](n) minus {1,n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A037279(n),n=1..80) ; # R. J. Mathar, Jul 23 2007

f[n_]:=If[PrimeQ[n],n,FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[n]]]]]]; Join[{1},Array[f,50,2]] (* Harvey P. Dale, Sep 24 2012 *)

More terms from Erich Friedman

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

A243535 Numbers whose list of divisors contains 2 distinct digits (in base 10).

Original entry on oeis.org

2, 3, 5, 7, 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

Offset: 1

Jaroslav Krizek, Jun 13 2014

Numbers k such that A037278(k), A176558(k) and A243360(k) contain 2 distinct digits.
Many of the composite terms are in A203897. - Charles R Greathouse IV, Sep 06 2016
Terms are either repdigit numbers (A010785) or contain only 1 and a single other digit. - Michael S. Branicky, Nov 16 2022

			121 is in the sequence because the list of divisors of 121, i.e., (1, 11, 121), contains 2 distinct digits (1, 2).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 4317 terms from Robert Israel)

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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1..10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=2;A(n)); Arrangement of column B]

dmax:= 6: # get all terms of <= dmax digits
Res:= {}:
for a in [0,$2..9] do
    S:= {0}:
    for d from 1 to dmax do
        S:= map(t -> (10*t+1,10*t+a), S);
        Res:= Res union select(filter, S)
    od
od:
sort(convert(Res,list)): # Robert Israel, Sep 05 2016

Select[Range[9000],Length[Union[Flatten[IntegerDigits/@Divisors[ #]]]] == 2&] (* Harvey P. Dale, Dec 14 2017 *)

isok(n) = vd = []; fordiv(n, d, vd = concat(vd, digits(d))); #Set(vd) == 2; \\ Michel Marcus, Jun 13 2014

from sympy import divisors
from itertools import count, islice, product
def ok(n):
    s = set("1"+str(n))
    if len(s) > 2: return False
    for d in divisors(n, generator=True):
        s |= set(str(d))
        if len(s) > 2: return False
    return len(s) == 2
def agen():
    yield from [2, 3, 5, 7]
    for d in count(2):
        s = set()
        for first, other in product("123456789", "0123456789"):
            for p in product(sorted(set(first+other)), repeat=d-1):
                if other not in p: continue
                t = int(first+"".join(p))
                if ok(t): s.add(t)
        yield from sorted(s)
print(list(islice(agen(), 52))) # Michael S. Branicky, Nov 16 2022

A243536 Numbers n such that list of divisors of n contains 3 distinct digits (in base 10).

Original entry on oeis.org

4, 9, 15, 23, 25, 29, 37, 39, 43, 44, 47, 53, 59, 67, 73, 79, 83, 89, 93, 95, 97, 99, 103, 107, 109, 111, 119, 122, 125, 127, 137, 139, 143, 149, 155, 157, 163, 167, 173, 179, 187, 193, 197, 202, 223, 227, 229, 233, 241, 242, 251, 271, 277, 281, 303, 317, 337

Offset: 1

Jaroslav Krizek, Jun 13 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 3 distinct digits.

			44 is in sequence because list of divisors of 44: (1, 2, 4, 11, 22, 44) contains 3 distinct digits (1, 2, 4).

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A095048, A037278, A176558, A243360.
Sequences of numbers n such that 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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1 …10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=3;A(n)); Arrangement of column B]

A243537 Numbers n such that list of divisors of n contains 4 distinct digits (in base 10).

Original entry on oeis.org

6, 8, 10, 14, 21, 26, 35, 49, 50, 51, 62, 65, 66, 82, 85, 88, 91, 110, 115, 117, 123, 133, 141, 142, 159, 165, 169, 183, 209, 213, 217, 221, 226, 231, 239, 244, 250, 253, 257, 262, 263, 269, 275, 283, 293, 295, 299, 307, 309, 319, 326, 333, 347, 349, 355, 359

Offset: 1

Jaroslav Krizek, Jun 13 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 4 distinct digits.

			49 is in sequence because divisors of 49: (1, 7, 49) contain 4 distinct digits (1, 4, 7, 9).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

Cf. A095048, A037278, A176558, A243360.
Sequences of numbers n such that 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)=4;A(n)); Arrangement of column B]

Select[Range[400],Length[Union[Flatten[IntegerDigits/@Divisors[#]]]] == 4&] (* Harvey P. Dale, Aug 22 2021 *)

A243538 Numbers n such that the list of divisors of n contains 5 distinct digits (in base 10).

Original entry on oeis.org

12, 16, 20, 27, 28, 34, 38, 45, 46, 57, 58, 69, 74, 75, 94, 100, 118, 124, 129, 132, 145, 153, 154, 161, 164, 166, 171, 175, 177, 178, 185, 194, 195, 205, 206, 214, 215, 218, 219, 220, 237, 254, 265, 273, 274, 279, 284, 287, 289, 291, 297, 298, 301, 302, 305

Offset: 1

Jaroslav Krizek, Jun 19 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 5 distinct digits.

			45 is in sequence because the list of divisors of 45: (1, 3, 5, 9, 15, 45) contains 5 distinct digits (1, 3, 4, 5, 9).

David A. Corneth, Table of n, a(n) for n = 1..10000

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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1...10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=5;A(n)); Arrangement of column B]

A243539 Numbers n such that the list of divisors of n contains 6 distinct digits (in base 10).

Original entry on oeis.org

18, 24, 30, 32, 40, 42, 48, 52, 63, 64, 81, 86, 87, 92, 98, 105, 106, 128, 130, 134, 146, 147, 148, 158, 176, 186, 188, 198, 200, 201, 203, 222, 235, 246, 247, 248, 249, 255, 259, 264, 278, 286, 310, 314, 322, 327, 328, 329, 330, 332, 334, 338, 346, 351, 357

Offset: 1

Jaroslav Krizek, Jun 19 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 6 distinct digits.

			48 is in sequence because the list of divisors of 48: (1, 2, 3, 4, 6, 8, 12, 16, 24, 48) contains 6 distinct digits (1, 2, 3, 4, 6, 8).

David A. Corneth, Table of n, a(n) for n = 1..10000

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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1...10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=6;A(n)); Arrangement of column B]

Select[Range[400],Length[Union[Flatten[IntegerDigits/@Divisors[#]]]]==6&] (* Harvey P. Dale, Apr 13 2025 *)

A243540 Numbers n such that the list of divisors of n contains 7 distinct digits (in base 10).

Original entry on oeis.org

36, 56, 60, 68, 70, 78, 80, 84, 96, 112, 116, 135, 136, 138, 150, 172, 184, 189, 190, 192, 196, 207, 212, 225, 230, 238, 243, 245, 256, 260, 261, 267, 268, 272, 285, 290, 292, 344, 345, 350, 358, 368, 384, 387, 388, 396, 400, 402, 418, 441, 444, 455, 459, 462

Offset: 1

Jaroslav Krizek, Jun 19 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 7 distinct digits.

			36 is in sequence because the list of divisors of 36: (1, 2, 3, 4, 6, 9, 12, 18, 36) contains 7 distinct digits (1, 2, 3, 4, 6, 8, 9).

David A. Corneth, Table of n, a(n) for n = 1..10000

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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1...10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=7;A(n)); Arrangement of column B]

A243541 Numbers n such that the list of divisors of n contains 8 distinct digits (in base 10).

Original entry on oeis.org

72, 76, 102, 104, 120, 126, 140, 144, 160, 168, 170, 182, 208, 210, 224, 232, 234, 236, 240, 258, 266, 276, 282, 288, 294, 296, 300, 308, 318, 320, 336, 352, 370, 372, 376, 416, 424, 430, 435, 436, 438, 448, 460, 464, 470, 476, 483, 494, 518, 520, 528, 536

Offset: 1

Jaroslav Krizek, Jun 19 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 8 distinct digits.

			72 is in sequence because the list of divisors of 72: (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) contains 8 distinct digits (1, 2, 3, 4, 6, 7, 8, 9).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harvey P. Dale)

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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1 …10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=8;A(n)); Arrangement of column B]

Select[Range[600],Length[Union[Flatten[IntegerDigits/@Divisors[#]]]]==8&] (* Harvey P. Dale, Jul 14 2016 *)

A243542 Numbers n such that the list of divisors of n contains 9 distinct digits (in base 10).

Original entry on oeis.org

54, 90, 114, 152, 156, 162, 174, 180, 204, 228, 252, 280, 315, 316, 340, 342, 348, 354, 356, 364, 378, 390, 392, 405, 408, 414, 420, 456, 468, 472, 474, 480, 486, 490, 510, 516, 522, 532, 534, 546, 552, 556, 560, 564, 576, 582, 584, 588, 592, 594, 600, 616

Offset: 1

Jaroslav Krizek, Jun 19 2014

Numbers n such that A037278(n), A176558(n) and A243360(n) contain 9 distinct digits.

			54 is in sequence because 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. 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 (the smallest number m whose list of divisors contains n distinct digits).

[Row n = 1 …10000; Column A: A(n) = A095048(n); Column B: B(n) = IF(A(n)=9;A(n)); Arrangement of column B]

A106708 a(n) is the concatenation of its nontrivial divisors.

Original entry on oeis.org

0, 0, 0, 2, 0, 23, 0, 24, 3, 25, 0, 2346, 0, 27, 35, 248, 0, 2369, 0, 24510, 37, 211, 0, 2346812, 5, 213, 39, 24714, 0, 23561015, 0, 24816, 311, 217, 57, 234691218, 0, 219, 313, 24581020, 0, 23671421, 0, 241122, 35915, 223, 0, 23468121624, 7, 251025, 317

Offset: 1

N. J. A. Sloane, Jul 20 2007

Klaus Brockhaus, Table of n, a(n) for n=1..5000

Cf. A037278, A120712, A037279, A131983 (records), A131984 (where records occur).

Cf. A027750, A010051, A037285, A037277, A163870.

a106708 1           = 0
a106708 n
   | a010051 n == 1 = 0
   | otherwise = read $ concat $ (map show) $ init $ tail $ a027750_row n
-- Reinhard Zumkeller, May 01 2012

A106708 := proc(n) local dvs ; if isprime(n) or n = 1 then 0; else dvs := [op(numtheory[divisors](n) minus {1,n} )] ; dvs := sort(dvs) ; cat(op(dvs)) ; fi ; end: seq(A106708(n),n=1..80) ; # R. J. Mathar, Aug 01 2007

Table[If[CompositeQ[n],FromDigits[Flatten[IntegerDigits/@Rest[ Most[ Divisors[ n]]]]],0],{n,60}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 22 2020 *)

{map(n) = local(d); d=divisors(n); if(#d<3, 0, d[1]=""; eval(concat(vecextract(d, concat("..", #d-1)))))}
for(n=1,51,print1(map(n),",")) /* Klaus Brockhaus, Aug 05 2007 */

from sympy import divisors
def a(n):
  nontrivial_divisors = [d for d in divisors(n)[1:-1]]
  if len(nontrivial_divisors) == 0: return 0
  else: return int("".join(str(d) for d in nontrivial_divisors))
print([a(n) for n in range(1, 52)]) # Michael S. Branicky, Dec 31 2020

a(n) = A037279(n) * A010051(n). - R. J. Mathar, Aug 01 2007

More terms from R. J. Mathar and Klaus Brockhaus, Aug 01 2007

Name edited by Michael S. Branicky, Dec 31 2020

cp's OEIS Frontend

A037279 If n is composite, replace n with the concatenation of its nontrivial divisors, otherwise a(n) = n.

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A243535 Numbers whose list of divisors contains 2 distinct digits (in base 10).

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A243536 Numbers n such that list of divisors of n contains 3 distinct digits (in base 10).

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A243537 Numbers n such that list of divisors of n contains 4 distinct digits (in base 10).

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A243538 Numbers n such that the list of divisors of n contains 5 distinct digits (in base 10).

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A243539 Numbers n such that the list of divisors of n contains 6 distinct digits (in base 10).

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A243540 Numbers n such that the list of divisors of n contains 7 distinct digits (in base 10).

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A243541 Numbers n such that the list of divisors of n contains 8 distinct digits (in base 10).

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