A327577 -id:A327577 - cp's OEIS frontend

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

Seiichi Manyama, Sep 16 2019

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

			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;

Seiichi Manyama, Rows n = 2..18, flattened
Wikipedia, Polydivisible number.

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

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)

A327571 Triangle T(n,k) read by rows giving the number of zeroless polydivisible numbers in base n that contains only "k" in the digits with 1 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 2, 2, 4, 2, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 4, 1, 2, 1, 2, 1, 3, 1, 2, 2, 4, 2, 2, 4, 2, 2, 1, 3, 1, 4, 1, 3, 1, 4, 1, 2, 2, 6, 2, 2, 6, 2, 2, 6, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 4, 4, 4, 6, 4, 4, 4, 4, 6, 4, 4

Offset: 2

Seiichi Manyama, Sep 17 2019

			n | zeroless polydivisible numbers with all digits the same in base n
--+------------------------------------------------------------------
2 | [1]
3 | [1, 11], [2, 22]
4 | [1], [2, 22, 222], [3]
So T(2,1) = 1, T(3,1) = 2, T(3,2) = 2, T(4,1) = 1, T(4,2) = 3, T(4,3) = 1.
Triangle begins:
n\k  | 1  2  3  4  5  6  7  8  9 10 11 12
-----+------------------------------------
   2 | 1;
   3 | 2, 2;
   4 | 1, 3, 1;
   5 | 2, 2, 4, 2;
   6 | 1, 2, 1, 2, 1;
   7 | 4, 4, 4, 4, 6, 4;
   8 | 1, 2, 1, 2, 1, 3, 1;
   9 | 2, 2, 4, 2, 2, 4, 2, 2;
  10 | 1, 3, 1, 4, 1, 3, 1, 4, 1;
  11 | 2, 2, 6, 2, 2, 6, 2, 2, 6, 2;
  12 | 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1;
  13 | 4, 4, 4, 4, 6, 4, 4, 4, 4, 6, 4, 4;

Seiichi Manyama, Rows n = 2..141, flattened
Wikipedia, Polydivisible number.

Row sums give A327577.

Cf. A071222, A327545.

def T(k, n)
  s = 0
  (0..n - 2).each{|i|
    s += k * n ** i
    return i if s % (i + 1) > 0
  }
  n - 1
end
def A327571(n)
  (2..n).map{|i| (1..i - 1).map{|j| T(j, i)}}.flatten
end
p A327571(10)

T(n,1) = T(n,n-1) = A071222(n-2).
T(n,1) <= T(n,k).
T(n,2*m) >= 2 for m >= 1.

cp's OEIS Frontend

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.

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