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 1-4 of 4 results.

A324020 Total number of zeroless polydivisible numbers in base n.

Original entry on oeis.org

1, 4, 9, 32, 45, 236, 330, 1108, 2157, 12740, 7713, 93710, 65602, 230342, 570128, 5007682, 2484863, 36896861, 16618196, 81481351, 266303823, 1991227852, 533069755, 7599786619, 13636829615, 35633175288, 43994413188, 796513902354, 121485971111, 5858898939564
Offset: 2

Views

Author

Seiichi Manyama, Sep 01 2019

Keywords

Examples

			n | polydivisible numbers in base n  | zeroless
--+----------------------------------+---------------
2 | [0, 1]                           | [1]
  | [10]                             |
--+----------------------------------+---------------
3 | [0, 1, 2]                        | [1, 2]
  | [11, 20, 22]                     | [11, 22]
  | [110, 200, 220]                  |
  | [1100, 2002, 2200]               |
  | [11002, 20022]                   |
  | [110020, 200220]                 |
--+----------------------------------+----------------
4 | [0, 1, 2, 3]                     | [1, 2, 3]
  | [10, 12, 20, 22, 30, 32]         | [12, 22, 32]
  | [102, 120, 123, 201,             | [123, 222, 321]
  |  222, 300, 303, 321]             |
  | [1020, 1200, 1230, 2010,         |
  |  2220, 3000, 3030, 3210]         |
  | [10202, 12001, 12303, 20102,     |
  |  22203, 30002, 32103]            |
  | [120012, 123030, 222030, 321030] |
  | [2220301]                        |

Links

Crossrefs

Cf. A181736, A271374, A324019, A324205.

Programs

  • Ruby
    def A(n)
  d = 0
  a = (1..n - 1).map{|i| [i]}
  cnt = n - 1
  while d < n - 2
    d += 1
    b = []
    a.each{|i|
      (1..n - 1).each{|j|
        m = i.clone + [j]
        if (0..d).inject(0){|s, k| s + m[k] * n ** (d - k)} % (d + 1) == 0
          b << m
          cnt += 1
        end
      }
    }
    a = b
  end
  cnt
end
def A324020(n)
  (2..n).map{|i| A(i)}
end
p A324020(10)

Formula

a(n) = Sum_{k=1..n-1} A324019(n,k).

Extensions

a(20)-a(31) from Bert Dobbelaere, Sep 14 2019

A327545 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that have k distinct digits with 1 <= k <= n-1.

Original entry on oeis.org

1, 4, 0, 5, 2, 2, 10, 14, 8, 0, 7, 14, 20, 2, 2, 26, 39, 84, 60, 27, 0, 11, 47, 108, 95, 63, 3, 3, 20, 101, 233, 369, 289, 79, 17, 0, 19, 86, 306, 475, 714, 409, 146, 1, 1, 32, 201, 979, 2048, 3581, 3474, 1925, 449, 51, 0, 17, 114, 507, 1273, 2224, 2239, 1074, 230, 35, 0, 0
Offset: 2

Views

Author

Seiichi Manyama, Sep 16 2019

Keywords

Comments

For k >= n there is no k-digit zeroless polydivisible number in base n.

Examples

			n | zeroless polydivisible numbers in base n
--+------------------------------------------
2 | [1]
3 | [1, 2, 11, 22]
4 | [1, 2, 3, 22, 222],  [12, 32], [123, 321]
So T(2,1) = 1, T(3,1) = 4, T(3,2) = 0, T(4,1) = 5, T(4,2) = 2, T(4,3) = 2.
Triangle begins:
n\k  |  1    2    3    4    5    6    7  8  9
-----+----------------------------------------
   2 |  1;
   3 |  4,   0;
   4 |  5,   2,   2;
   5 | 10,  14,   8,   0;
   6 |  7,  14,  20,   2,   2;
   7 | 26,  39,  84,  60,  27,   0;
   8 | 11,  47, 108,  95,  63,   3,   3;
   9 | 20, 101, 233, 369, 289,  79,  17, 0;
  10 | 19,  86, 306, 475, 714, 409, 146, 1, 1;

Links

Crossrefs

Row sums give A324020.
T(2*n,2*n-1) gives A181736.
T(n,1) gives A327577.
Cf. A324019, A324205.

Programs

  • Ruby
    def A(n)
  d = 0
  a = (1..n - 1).map{|i| [i]}
  ary = [n - 1] + Array.new(n - 2, 0)
  while d < n - 2
    d += 1
    b = []
    a.each{|i|
      (1..n - 1).each{|j|
        m = i.clone + [j]
        if (0..d).inject(0){|s, k| s + m[k] * n ** (d - k)} % (d + 1) == 0
          b << m
          ary[m.uniq.size - 1] += 1
        end
      }
    }
    a = b
  end
  ary
end
def A327545(n)
  (2..n).map{|i| A(i)}.flatten
end
p A327545(10)

A324205 The largest zeroless polydivisible number in base n (written in base 10).

Original entry on oeis.org

1, 8, 57, 616, 7465, 117636, 2054451, 42912896, 987654564, 25915636800, 736867916290, 23297940244152, 789024560946557, 29174313555611252, 1147797409031506920, 48649494479090875520, 2178349008754768757872, 104127216885225393371514
Offset: 2

Views

Author

Seiichi Manyama, Sep 02 2019

Keywords

Examples

			a(3) = (22)_3 = 8.
a(4) = (321)_4 = 57.
a(5) = (4431)_5 = 616.
a(6) = (54321)_6 = 7465.
a(7) = (666651)_7 = 117636.
a(8) = (7654463)_8 = 2054451.
a(9) = (88665375)_9 = 42912896.
a(10) = 987654564.
a(11) = (AA99779559)_11 = 25915636800.
a(12) = (BA9876A836A)_12 = 736867916290.

Links

Crossrefs

Cf. A109032, A324019, A324020.

Formula

a(n) is A324020(n)-th zeroless polydivisible number in base n (written in base 10).

A324022 Zeroless "magic" numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38, 42, 44, 46, 48, 52, 54, 56, 58, 62, 64, 66, 68, 72, 74, 76, 78, 82, 84, 86, 88, 92, 94, 96, 98, 123, 126, 129, 141, 144, 147, 162, 165, 168, 183, 186, 189, 222, 225, 228, 243, 246, 249, 261, 264, 267, 282, 285
Offset: 1

Views

Author

Seiichi Manyama, Sep 01 2019

Keywords

Comments

Intersection of A052382 and A144688.
The 2157th and largest term is the 9-digit number 987654564.

Examples

			98        = 2 * 49,
987       = 3 * 329,
9876      = 4 * 2469,
98765     = 5 * 19753,
987654    = 6 * 164609,
9876545   = 7 * 1410935,
98765456  = 8 * 12345682,
987654564 = 9 * 109739396.
So 987654564 is a term.

Links

Crossrefs

Cf. A052382, A144688, A324019, A324020.
Showing 1-4 of 4 results.

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