A327545 -id:A327545 - cp's OEIS frontend

A324019 Triangle T(n,k) read by rows giving the number of k-digit zeroless polydivisible numbers in base n with 1 <= k <= n-1.

1, 2, 2, 3, 3, 3, 4, 8, 10, 10, 5, 10, 10, 10, 10, 6, 18, 36, 52, 62, 62, 7, 21, 49, 49, 68, 68, 68, 8, 32, 64, 128, 205, 205, 233, 233, 9, 36, 108, 216, 216, 324, 416, 416, 416, 10, 50, 166, 416, 832, 1382, 1967, 2465, 2726, 2726, 11, 55, 165, 330, 726, 726, 1140, 1140, 1140, 1140, 1140

Offset: 2

Seiichi Manyama, Sep 01 2019

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

			Triangle begins:
n\k  | 1   2    3    4    5    6    7    8    9
-----+------------------------------------------
   2 | 1;
   3 | 2,  2;
   4 | 3,  3,   3;
   5 | 4,  8,  10,  10;
   6 | 5, 10,  10,  10,  10;
   7 | 6, 18,  36,  52,  62,  62;
   8 | 7, 21,  49,  49,  68,  68,  68;
   9 | 8, 32,  64, 128, 205, 205, 233, 233;
  10 | 9, 36, 108, 216, 216, 324, 416, 416, 416;

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

Row sums give A324020.

Cf. A271373, A327545.

A327577 Numbers of zeroless polydivisible numbers with all digits the same in base n.

Original entry on oeis.org

1, 4, 5, 10, 7, 26, 11, 20, 19, 32, 17, 52, 22, 36, 38, 44, 28, 78, 32, 64, 49, 60, 38, 104, 47, 70, 61, 78, 49, 196, 53, 88, 75, 94, 66, 162, 64, 104, 88, 134, 70, 216, 74, 120, 123, 128, 80, 214, 85, 168, 117, 144, 91, 240, 103, 162, 131, 160, 101, 392, 108, 172, 152, 178, 122, 296

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 a(2) = 1, a(3) = 4, a(4) = 5.

Seiichi Manyama, Table of n, a(n) for n = 2..10000
Wikipedia, Polydivisible number.

Cf. A327545, A327571.

a(n) = A327545(n,1).

a(n) = Sum_{k=1..n-1} A327571(n,k).

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

A324019 Triangle T(n,k) read by rows giving the number of k-digit zeroless polydivisible numbers in base n with 1 <= k <= n-1.

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A327577 Numbers of zeroless polydivisible numbers with all digits the same in base n.

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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.

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