cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Reinhard Zumkeller, May 28 2004

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Haskell
    import Data.List (elemIndices)
a095050 n = a095050_list !! (n-1)
a095050_list = map (+ 1) $ elemIndices 10 $ map a095048 [1..]
-- Reinhard Zumkeller, Feb 05 2012
  • Maple
    q:= n-> is({$0..9}=map(x-> convert(x, base, 10)[], numtheory[divisors](n))):
select(q, [$1..2000])[];  # Alois P. Heinz, Oct 28 2021
  • Mathematica
    Select[Range@2000, 1+Union@@IntegerDigits@Divisors@# == Range@10 &] (* Hans Rudolf Widmer, Oct 28 2021 *)
  • PARI
    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
  • Python
    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

Formula

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

Views

Author

Reinhard Zumkeller, May 28 2004

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A095049, A095050, A043537.
Cf. A027750, A059436.

Programs

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

A062634 Numbers k such that every divisor of k contains the digit 1.

Original entry on oeis.org

1, 11, 13, 17, 19, 31, 41, 61, 71, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 221, 241, 251, 271, 281, 311, 313, 317, 331, 341, 361, 401, 419, 421, 431, 451, 461, 491, 521
Offset: 1

Views

Author

Erich Friedman, Jul 04 2001

Keywords

Comments

First composite term is 121. All powers of 11 are in the sequence. - Alonso del Arte, Sep 29 2013

Examples

			143 has divisors 1, 11, 13 and 143, all of which contain the digit 1.

Links

Crossrefs

Cf. A027750, subsequence of A011531; A206159 and A208270 are subsequences.
Cf. A001020 (powers of 11).

Programs

  • Haskell
    a062634 n = a062634_list !! (n-1)
a062634_list = filter
   (and . map ((elem '1') . show) . a027750_row) a011531_list
-- Reinhard Zumkeller, Feb 05 2012
  • Maple
    q:= n-> andmap(x-> 1 in convert(x, base, 10), numtheory[divisors](n)):
select(q, [$1..1000])[];  # Alois P. Heinz, May 09 2022
  • Mathematica
    fQ[n_, dgt_] := Union[ MemberQ[#, dgt] & /@ IntegerDigits@ Rest@ Divisors@ n][[1]]; Select[ Range[2, 525], fQ[#, 1] &] (* Robert G. Wilson v, Jun 11 2014 *)
  • PARI
    isok(m) = fordiv(m, d, if (! #select(x->(x==1), digits(d)), return(0))); return(1); \\ Michel Marcus, May 09 2022

Extensions

Offset corrected by Reinhard Zumkeller, Feb 05 2012
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.