A324020 Total number of zeroless polydivisible numbers in base n.

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

2,2

Links

Table of n, a(n) for n=2..31.

Wikipedia, Polydivisible number.

Formula

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

Example

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]                        |

Prog

(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)

Crossrefs

Cf. A181736, A271374, A324019, A324205.

Sequence in context: A338576 A270618 A270634 * A265645 A005985 A322780

Adjacent sequences: A324017 A324018 A324019 * A324021 A324022 A324023

Keyword

nonn,base

Author

Seiichi Manyama, Sep 01 2019

Extensions

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

Status

approved