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.

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

2,2

Comments

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

Links

Seiichi Manyama, Rows n = 2..18, flattened

Wikipedia, Polydivisible number.

Example

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;

Prog

(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)

Crossrefs

Row sums give A324020.

T(2*n,2*n-1) gives A181736.

T(n,1) gives A327577.

Cf. A324019, A324205.

Sequence in context: A309215 A309216 A083745 * A306072 A185199 A201196

Adjacent sequences: A327542 A327543 A327544 * A327546 A327547 A327548

Keyword

nonn,base,tabl

Author

Seiichi Manyama, Sep 16 2019

Status

approved