A059436 -id:A059436 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

A345390 Numbers whose set of divisors contains every digit at least twice.

Original entry on oeis.org

540, 720, 760, 810, 918, 1080, 1140, 1170, 1260, 1404, 1440, 1512, 1520, 1530, 1560, 1620, 1740, 1800, 1820, 1824, 1836, 1872, 1890, 1908, 1960, 2016, 2028, 2052, 2070, 2072, 2088, 2106, 2112, 2124, 2142, 2156, 2160, 2184, 2208, 2280, 2340, 2380, 2430, 2508, 2520

Offset: 1

Tanya Khovanova, Jun 17 2021

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

			The divisors of 918 are 1, 2, 3, 6, 9, 17, 18, 27, 34, 51, 54, 102, 153, 306, 459, and 918. Every digit appears at least twice. Thus, 918 is in this sequence.

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

Cf. A059436, A095050.

q:= n-> (p-> is(min(seq(coeff(p, x, j), j=0..9))>1))(add(x^i, i=
     map(d-> convert(d, base, 10)[], [numtheory[divisors](n)[]]))):
select(q, [$10..2600])[];  # Alois P. Heinz, Apr 21 2022

Select[Range[3000], Length[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] == 10 && Min[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] > 1 &]

from sympy import divisors
def ok(n):
    digits_used = {d:0 for d in "0123456789"}
    for div in divisors(n, generator=True):
        for d in str(div): digits_used[d] += 1
        if all(digits_used[d] > 1 for d in "0123456789"): return True
    return False
print([k for k in range(2521) if ok(k)]) # Michael S. Branicky, Jun 25 2022

A175507 Numbers such that each digit from 0 to 9 appears at least 7 times in the digits of their divisors.

Original entry on oeis.org

7980, 8190, 9360, 10920, 11760, 11880, 12870, 13230, 13860, 14820, 15960, 16380, 16740, 17640, 17940, 18216, 18270, 18360, 18720, 18810, 18900, 19040, 19080, 19140, 19656, 19740, 20196, 20580, 20790, 20880, 21168, 21560, 21840, 22176

Offset: 1

R. J. Mathar, Jun 02 2010

			5460 is not in the sequence because the divisors, 1, 2, 3, 4, 5, 6, 7, 10, 12, 13, 14, 15, 20, 21, 26, 28, 30, 35, 39, 42, 52, 60, 65, 70, 78, 84, 91, 105, 130, 140, 156, 182, 195, 210, 260, 273, 364, 390, 420, 455, 546, 780, 910, 1092, 1365, 1820, 2730, 5460, contain the digit 7 only 6 times (namely once in 7, 70, 78, 273, 780 and 2730), which is not enough.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Erich Friedman, What's Special About This Number? (See entry 7980.)

Cf. A059436.

fQ[n_] := Block[{s = Transpose@ Tally[ Sort[ Flatten[ IntegerDigits@# & /@ Divisors@ n]]]}, First@ s == {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} && Min@ Last@ s > 6]; k = 1; lst = {}; While[k < 22319, If[ fQ@k, AppendTo[lst, k]]; k++ ]; lst (* Robert G. Wilson v, Jul 31 2010 *)

More terms from Robert G. Wilson v, Jul 31 2010

A353066 Numbers whose set of divisors contains every digit at least three times.

Original entry on oeis.org

1140, 1890, 2280, 2340, 2610, 2660, 2700, 2808, 2880, 2940, 2970, 3420, 3480, 3510, 3540, 3600, 3654, 3672, 3780, 3870, 3920, 3952, 3990, 4032, 4140, 4320, 4368, 4380, 4410, 4560, 4590, 4680, 4740, 4760, 4770, 4860, 4896, 4940, 4950, 5130, 5220, 5320, 5400, 5454

Offset: 1

Tanya Khovanova, Apr 21 2022

Every multiple of a term is also a term.

			The divisors of 1140 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 19, 20, 30, 38, 57, 60, 76, 95, 114, 190, 228, 285, 380, 570, 1140. Digits tally from 0 to 9: 8, 10, 6, 4, 3, 6, 3, 3, 4, 3. The minimum is 3, thus, 1140 is in this sequence.

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

Cf. A059436 (at least n times).

Subsequence of A095050 (at least once) and of A345390 (at least twice).

q:= n-> (p-> is(min(seq(coeff(p, x, j), j=0..9))>2))(add(x^i, i=
     map(d-> convert(d, base, 10)[], [numtheory[divisors](n)[]]))):
select(q, [$10..5555])[];  # Alois P. Heinz, Apr 21 2022

Select[Range[10000], Length[Tally[Flatten[IntegerDigits[Divisors[#]]]]] == 10 && Min[Transpose[Tally[Flatten[IntegerDigits[Divisors[#]]]]][[2]]] >= 3 &]

upto(n) = { my(v = vector(n, i, -1)); for(i = 1, n, if(v[i] == -1, c = is(i); if(c == 1, v[i] = 1; for(j = 1, n\i, v[i*j] = 1; ) , v[i] = 0 ) ) ); Vec(select(x->x==1,v,1)) }
is(n) = { my(v = vector(10, i, 3), d = divisors(n), todo = 30, i, j); for(i = 1, #d, dd = digits(d[i]); for(j = 1, #dd, if(v[dd[j]+1] > 0, v[dd[j]+1]--; todo--; if(todo <= 0, return(1) ) ) ) ); 0 } \\ David A. Corneth, Jul 11 2022

from sympy import divisors
def ok(n):
    counts = [0]*10
    for d in divisors(n, generator=True):
        for di in str(d): counts[int(di)] += 1
        if min(counts) > 2: return True
    return False
print([k for k in range(5455) if ok(k)]) # Michael S. Branicky, Apr 21 2022

A273094 a(n) is the smallest number whose divisors contain each 0..9 digit exactly n times.

Original entry on oeis.org

203457869, 206893558, 507083396, 506815954, 102668478970, 895233580, 26475394180, 887692930, 10708845258, 13306408052, 155503137452, 963213572, 803503960576, 40349550036, 203264657940

Offset: 1

Giovanni Resta, May 15 2016

We also have a(18)=18174907880, a(19)=81418065258, a(20)=257678968520, and a(23)=529539876740. The missing terms are all greater than 10^12.

			a(1)=203457869, whose divisors are 1 and 203457869 itself. a(2)=206893558, whose divisors, i.e., 1, 2, 103446779, and 206893558, contain each digit 2 times.

Cf. A038564, A059436, A243363.

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

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

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A345390 Numbers whose set of divisors contains every digit at least twice.

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A175507 Numbers such that each digit from 0 to 9 appears at least 7 times in the digits of their divisors.

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A353066 Numbers whose set of divisors contains every digit at least three times.

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A273094 a(n) is the smallest number whose divisors contain each 0..9 digit exactly n times.

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