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

Author

Seiichi Manyama, Sep 01 2019

Keywords

• nonn
• base

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

• Wikipedia, Polydivisible number.

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