A324020
Total number of zeroless polydivisible numbers in base n.
Original entry on oeis.org
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
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] |
A334318
Number T(n,k) of integers in base n having exactly k distinct digits such that the number formed by the consecutive subsequence of the initial j digits is divisible by j for all j in {1,...,k}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 2, 1, 3, 1, 0, 4, 5, 5, 2, 5, 6, 6, 1, 0, 6, 13, 18, 8, 7, 2, 7, 15, 33, 34, 16, 7, 0, 8, 25, 50, 58, 52, 21, 8, 3, 9, 28, 67, 98, 101, 57, 30, 7, 0, 10, 41, 115, 168, 220, 88, 51, 9, 4, 1, 11, 45, 134, 275, 398, 315, 220, 126, 32, 10, 0, 12, 61, 206, 428, 690, 568, 503, 158, 32, 5, 1, 0
Offset: 1
T(4,3) = 5: 102, 120, 201, 123, 321 (written in base 4):
T(7,2) = 15: 13, 15, 20, 24, 26, 31, 35, 40, 42, 46, 51, 53, 60, 62, 64 (written in base 7)
T(10,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
T(10,10) = 1: 3816547290.
Triangle T(n,k) begins:
1;
2, 1;
3, 1, 0;
4, 5, 5, 2;
5, 6, 6, 1, 0;
6, 13, 18, 8, 7, 2;
7, 15, 33, 34, 16, 7, 0;
8, 25, 50, 58, 52, 21, 8, 3;
9, 28, 67, 98, 101, 57, 30, 7, 0;
10, 41, 115, 168, 220, 88, 51, 9, 4, 1;
11, 45, 134, 275, 398, 315, 220, 126, 32, 10, 0;
12, 61, 206, 428, 690, 568, 503, 158, 32, 5, 1, 0;
...
Bisection of main diagonal (even part) gives
A181736.
-
b:= proc(n, s, w) option remember; `if`(s={}, 0, (k-> add((t->
`if`(t=0, x, `if`(irem(t, k)=0, b(n, s minus {j}, t)
+x^k, 0)))(w*n+j), j=s)))(1+n-nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, {$0..n-1}, 0)):
seq(T(n), n=1..14);
-
b[n_, s_, w_] := b[n, s, w] = If[s == {}, 0, With[{k = 1+n-Length[s]}, Sum[With[{t = w*n + j}, If[t == 0, x, If[Mod[t, k] == 0, b[n, s ~Complement~ {j}, t] + x^k, 0]]], {j, s}]]];
T[n_] := PadRight[CoefficientList[b[n, Range[0, n-1], 0]/x, x], n];
Array[T, 14] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)
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.
Original entry on oeis.org
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
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;
-
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)
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