A156071 -id:A156071 - cp's OEIS frontend

A156069 a(n) = A156071(n)/n.

Original entry on oeis.org

3, 19, 127, 954, 7633, 63609, 545221, 4770684, 42406081

Offset: 1

Alexander R. Povolotsky, Sep 25 2009

Blaine, How about a math puzzle?
Albert Franck, Puzzles, see item 7.

Cf. A156071, A163574.

New name from Michel Marcus, Dec 01 2013

A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

Original entry on oeis.org

1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 1

Offset: 0

Gus Wiseman, Jun 08 2018

A sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. It is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

			Sequence of sets of n-digit numbers that are weakly polydivisible and strictly pandigital is (with A = 10):
  {0}
  {1}
  {12}
  {123,321}
  {}
  {}
  {123654,321654}
  {}
  {38165472}
  {381654729}
  {381654729A}

Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305715.

A305712 Polydivisible nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

Original entry on oeis.org

0, 10, 102, 120, 201, 1020, 1200, 2012, 10200, 12000, 12320, 20120, 32120, 102000, 120000, 123204, 321204, 1024023, 1200003, 1232042, 1444023, 2220001, 3212041, 10240232, 12000032, 12320424, 14440232, 32125240, 50165432

Offset: 0

Gus Wiseman, Jun 08 2018

A number with decimal digit sequence {q_1, ..., q_k} is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.

Wikipedia, Polydivisible number

Cf. A000670, A030299, A050289, A144688, A156069, A156071, A240763, A305701, A305714, A305715.

polyQ[q_]:=And@@Table[Divisible[FromDigits[Take[q,k]],k],{k,Length[q]}];
normseqs[n_]:=Join@@Permutations/@Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Sort[FromDigits/@Join@@Table[Select[normseqs[n]-1,First[#]>0&&polyQ[#]&],{n,8}]]

A305715 Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 1, 1, 2, 3, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 8, 1, 6, 5, 4, 7, 2, 3, 8, 1, 6, 5, 4, 7, 2, 9, 3, 8, 1, 6, 5, 4, 7, 2, 9, 10

Offset: 1

Gus Wiseman, Jun 08 2018

A positive integer sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. It is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

			Triangle is:
  {1}
  {1,2}
  {1,2,3}
  {3,2,1}
  {1,2,3,6,5,4}
  {3,2,1,6,5,4}
  {3,8,1,6,5,4,7,2}
  {3,8,1,6,5,4,7,2,9}
  {3,8,1,6,5,4,7,2,9,10}

Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.

Wikipedia, Polydivisible number

Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305714 (row lengths).

polyQ[q_]:=And@@Table[Divisible[FromDigits[Take[q,k]],k],{k,Length[q]}];
Flatten[Table[Select[Permutations[Range[n]],polyQ],{n,8}]]

A324016 N-digit substring of 81654327 taken from the left.

Original entry on oeis.org

8, 81, 816, 8165, 81654, 816543, 8165432, 81654327

Offset: 1

Seiichi Manyama, Sep 01 2019

A sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. Consider a strictly pandigital sequence q such that Sum_{i=1..m} 10^(m - i) * q_i is a multiple of m+1 for all 1 <= m <= k. Except for 21 and 81654327, there is no such sequence for k <= 30.

			8        = 2 * 4.
81       = 3 * 27.
816      = 4 * 204.
8165     = 5 * 1633.
81654    = 6 * 13609.
816543   = 7 * 116649.
8165432  = 8 * 1020679.
81654327 = 9 * 9072703.

Cf. A111456, A156071, A305714.

cp's OEIS Frontend

A156069 a(n) = A156071(n)/n.

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A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

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A305712 Polydivisible nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

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A305715 Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.

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Mathematica

A324016 N-digit substring of 81654327 taken from the left.

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