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.

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

Author

Seiichi Manyama, Sep 17 2019

Keywords

• nonn
• tabl
• base

Examples

			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;
		

Links

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

Crossrefs

Row sums give A327577.

Cf. A071222, A327545.

Programs

Ruby

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)

Formula

T(n,1) = T(n,n-1) = A071222(n-2).

T(n,1) <= T(n,k).

T(n,2*m) >= 2 for m >= 1.