A095050 -id:A095050 - cp's OEIS frontend

A095048 Number of distinct digits needed to write all positive divisors of n in decimal representation.

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

Offset: 1

Reinhard Zumkeller, May 28 2004

a(n) <= 10, a(A095050(n)) = 10.
a(A206159(n)) <= 2. - Reinhard Zumkeller, Feb 05 2012
Almost all (in the sense of natural density) terms of this sequence are equal to 10. - Charles R Greathouse IV, Nov 16 2022

			Set of divisors of n=10: {1,2,5,10}, therefore a(10) = #{0,1,2,5} = 4.
Set of divisors of n=16: {1,2,4,8,16}, therefore a(16)=#{1,2,4,6,8} = 5.

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

Cf. A095049, A095050, A043537.

Cf. A027750, A059436.

import Data.List (group, sort)
a095048 = length . group . sort . concatMap show . a027750_row
-- Reinhard Zumkeller, Feb 05 2012

A095048 := proc(n)
    local digset ;
    digset := {} ;
    for d in numtheory[divisors](n) do
        digset := digset union convert(convert(d,base,10),set) ;
    end do:
    nops(digset) ;
end proc:
seq(A095048(n),n=1..80) ; # R. J. Mathar, May 13 2022

a(n) = my(d = divisors(n), s = 0); for(i = 1, #d, v = digits(d[i]); for(j = 1, #v, s = bitor(s, 1<David A. Corneth, Nov 16 2022

from sympy import divisors
def a(n):
    s = set("1"+str(n))
    if len(s) == 10: return 10
    for d in divisors(n, generator=True):
        s |= set(str(d))
        if len(s) == 10: return 10
    return len(s)
print([a(n) for n in range(1, 99)]) # Michael S. Branicky, 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).

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]

cp's OEIS Frontend

A095048 Number of distinct digits needed to write all positive divisors of n in decimal representation.

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

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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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