cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Seiichi Manyama, Sep 01 2019

Keywords

Examples

			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]                        |

Links

Crossrefs

Cf. A181736, A271374, A324019, A324205.

Programs

  • Ruby
    def A(n)
  d = 0
  a = (1..n - 1).map{|i| [i]}
  cnt = n - 1
  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
          cnt += 1
        end
      }
    }
    a = b
  end
  cnt
end
def A324020(n)
  (2..n).map{|i| A(i)}
end
p A324020(10)

Formula

a(n) = Sum_{k=1..n-1} A324019(n,k).

Extensions

a(20)-a(31) from Bert Dobbelaere, Sep 14 2019

A271373 Triangle T(n,k) read by rows giving the number of k-digit polydivisible numbers (see A144688) in base n with 1 <= k <= A109783(n).

Original entry on oeis.org

2, 1, 3, 3, 3, 3, 2, 2, 4, 6, 8, 8, 7, 4, 1, 5, 10, 17, 21, 21, 21, 13, 10, 6, 4, 6, 15, 30, 45, 54, 54, 49, 46, 21, 3, 1, 7, 21, 49, 87, 121, 145, 145, 145, 121, 92, 56, 33, 20, 14, 7, 3, 1, 1, 8, 28, 74, 148, 238, 324, 367, 367, 320, 258, 188, 122, 69, 37, 12, 6, 3
Offset: 2

Views

Author

Martin Renner, Apr 05 2016

Keywords

Examples

			The triangle begins
n\k 1  2  3  4  5  6  7  8  9 10 ...
2:  2  1
3:  3  3  3  3  2  2
4:  4  6  8  8  7  4  1
5:  5 10 17 21 21 21 13 10  6  4
...

Links

Crossrefs

Cf. A109783 (row lengths), A143671 (row n=10), A144688, A271374 (row sums).

Programs

  • Maple
    b:=10; # Base
P:={seq(i,i=1..b-1)}: # Polydivisible numbers
M:=[nops(P)+1]: # Number of k-digit polydivisible numbers
for i from 2 while nops(P)>0 do
  Q:={}:
  for n from 1 to nops(P) do
    for j from 0 to b-1 do
      if P[n]*b+j mod i = 0 then Q:={op(Q),P[n]*b+j}: fi:
    od:
  od:
  M:=[op(M),nops(Q)]:
  P:=Q;
od:
T||b:=op(M[1..nops(M)-1]); # Table row T(n,k) for n = b

Formula

T(n,k) ~ (n-1)*n^(k-1)/k!
T(10,k) = A143671(k), 1 <= k <= 25.

Extensions

Rows n=17 to n=25 added to b-file by Max Alekseyev, Sep 11 2021

A380359 a(n) is the number of integers in base n such that all the integers given by their first k digits are divisible by k and which cannot be extended further.

Original entry on oeis.org

1, 3, 8, 21, 54, 145, 367, 1039, 2492, 6709, 16799, 46610, 95597, 368134, 831886, 2245056, 6084180, 15798495, 41456343, 119786906, 292818176, 788255058, 2061079489, 5753392327, 14984432350
Offset: 2

Views

Author

Inigo Quilez, Jan 22 2025

Keywords

Examples

			a(10)=2492 because from all A271374(10)=20457 polydivisible numbers, only 2492 cannot be further expanded into a larger polydivisible number. One such number is 4836545640368400: 4 is divisible by 1, 48 is divisible by 2, 483 is divisible by 3, 4836 is divisible by 4, and so on until 4836545640368400 which is divisible by 16; but one cannot extend it further since no digit (0 to 9) appended to 4836545640368400 would result in a number divisible by k=17.

Links

Crossrefs

Cf. A271374, A109783, A109032.
Showing 1-3 of 3 results.

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