A144688 -id:A144688 - cp's OEIS frontend

A143671 Number of terms in A144688 of length n.

10, 45, 150, 375, 750, 1200, 1713, 2227, 2492, 2492, 2225, 2041, 1575, 1132, 770, 571, 335, 180, 90, 44, 18, 12, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Zak Seidov and N. J. A. Sloane, Feb 08 2009

Since any 5-digit member of A144688 must end in 0 or 5, a(5) = 2*a(4).
Similarly a(n) <= 2*a(n-1) for n = 6, 7, 8, 9, a(10) = a(9), and a(n) <= a(n-1) for n >= 11.
a(n) = 0 for n > 25.

Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.

Cf. A144688.

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

Martin Renner, Apr 05 2016

			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
...

Max Alekseyev, Table of n, a(n) for n = 2..810
Wikipedia, Polydivisible number.

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

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

T(n,k) ~ (n-1)*n^(k-1)/k!

T(10,k) = A143671(k), 1 <= k <= 25.

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

A156071 Concatenation chain arising in A156069.

Original entry on oeis.org

3, 38, 381, 3816, 38165, 381654, 3816547, 38165472, 381654729

Offset: 1

Alexander R. Povolotsky, Sep 25 2009

a(9) is a zeroless pandigital number in base 10, with 9 digits such that every k-digit substring ( 1 <= k <= 9 ) taken from the left, is divisible by k (see A163574). - Michel Marcus, Dec 01 2013

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

Blaine, How about a math puzzle?
Albert Franck, Puzzles, see item 7.
Wikipedia, Polydivisible number
Jim Wilson, Statement of Nine-Digit Number puzzle

Cf. A010784, A030299, A050289, A143671, A144688, A156069, A158242, A163574.

A214437 Least numbers whose groups of 2,3,...,n digits taken from the left are divisible by 2,3,...,n.

Original entry on oeis.org

1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800

Offset: 1

Robin Garcia, Jul 17 2012

The first 11 terms of the sequence are coincident with A078282.
a(6) is formed with 66,7 % zeros;  A(5) with 60 %; a(7) with 57,1 %; a(4), a(8), a(10) and a(20) with 50 %.
a(n) is the first term of A144688 with n digits, except that A144688 includes zero as first term. - Franklin T. Adams-Watters, Jul 18 2012
There are 25 terms in the sequence; the 25-digit number 3608528850368400786036725 is the last number to satisfy the requirements. - Shyam Sunder Gupta, Aug 04 2013

			a(6) = 102000 because 10, 102, 1020, 10200 and 102000 are divisible by 2, 3, 4, 5 and 6.
There are nine one-digit numbers that are divisible by 1; the smallest is 1, so a(1)=1.
For two-digit numbers, the second digit must be even, i.e., 0,2,4,6,8 to make it divisible by 2, which gives 10 as the smallest number to satisfy the requirement, so a(2)=10. - _Shyam Sunder Gupta_, Aug 04 2013

Shyam Sunder Gupta, Table of n, a(n) for n = 1..25
Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.

Cf. A078282, A158242, A144688.

a=Table[j, {j, 9}]; r=2; t={};
While[!a == {}, n=Length[a]; nmin=Last[a]; k=1; b={};
While[!k>n, z0=a[[k]]; Do[z=10*z0+j; If[Mod[z, r]==0, b=Append[b, z]], {j, 0, 9}]; k++]; AppendTo[t, nmin]; a=b; r++]; t (* Shyam Sunder Gupta, Aug 04 2013 *)

A305701 Nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

Original entry on oeis.org

0, 10, 100, 101, 102, 110, 120, 201, 210, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1023, 1032, 1100, 1101, 1102, 1110, 1120, 1200, 1201, 1202, 1203, 1210, 1220, 1230, 1302, 1320, 2001, 2010, 2011, 2012, 2013, 2021, 2031, 2100, 2101, 2102, 2103

Offset: 1

Gus Wiseman, Jun 08 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000670, A030299, A050289, A144688, A240763, A305712, A305714, A305715.

filter:= proc(n) local L;
  L:= convert(convert(n,base,10),set);
  L = {$0..max(L)}
end proc:
select(filter, [$0..3000]); # Robert Israel, Jun 10 2018

Select[Range[0,10000],Union[IntegerDigits[#]]==Range[0,Max[IntegerDigits[#]]]&]

isok(n) = if (n==0, return (1)); my(d=Set(digits(n))); (vecmin(d) == 0) && (vecmax(d) == #d - 1); \\ Michel Marcus, Jul 05 2018

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.

A082399 a(1) = 1; thereafter, a(n) is the smallest nonnegative number such that the number Sum_{i=1..n} a(i)*10^(n-i) is divisible by n.

Original entry on oeis.org

1, 0, 2, 0, 0, 0, 5, 6, 4, 0, 5, 10, 2, 2, 5, 6, 16, 8, 14, 0, 7, 18, 19, 2, 5, 10, 6, 0, 25, 20, 2, 20, 17, 12, 20, 28, 13, 4, 13, 30, 16, 20, 36, 4, 35, 28, 28, 16, 29, 10, 39, 14, 12, 4, 50, 20, 14, 24, 7, 50, 14, 54, 55, 18, 10, 44, 62, 52, 63, 50, 7, 18, 6, 62, 55, 54, 54, 54, 35, 10

Offset: 1

N. J. A. Sloane, Feb 08 2009

nonn

Suggested by studying A144688. If all a(n) had turned out to be in the range 0 to 9 then this sequence would have produced a counterexample to the assertion that A144688 is finite.

The old entry with this A-number was a duplicate of A080825.

			After we have the first 11 terms, 1,0,2,0,0,0,5,6,4,0,5, the next number x must be chosen so that 102000564050 + x is divisible by 12; this implies that x = 10.

Paul Tek, Table of n, a(n) for n = 1..10000

See A051883 for another version. Cf. A144688.

M:=80; a[1]:=1; N:=1;
for n from 2 to M do
N:=10*N; t2:=N mod n;
if t2 = 0 then a[n]:=0; else a[n]:=n-t2; fi;
N:=N+a[n]; od: [seq(a[n],n=1..M)];

A271374 Total number of polydivisible numbers in base n.

Original entry on oeis.org

3, 16, 38, 128, 324, 1068, 2569, 8381, 20457, 58174, 148059, 441493, 916146, 3722968, 8407790, 23909586, 64576509, 178009925, 466027279, 1409607602, 3507905894, 9694292108, 25391646456, 73838562312, 191793924162, 550333004128

Offset: 2

Martin Renner, Apr 05 2016

			There are a(10) = 20457 polydivisible numbers in base 10, which are listed in A144688.

Wikipedia, Polydivisible number.

Row sums of A271373.

Cf. A109032, A109783, A144688.

a(n) = Sum_{k=1..A109783(n)} A271373(n,k).

a(n) ~ (n-1)*(exp(n)-1)/n.

a(16) from Seiichi Manyama, Sep 01 2019
a(17)-a(18) from Seiichi Manyama, Sep 02 2019
a(19)-a(27) from Max Alekseyev, Sep 08 2021

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}]]

cp's OEIS Frontend

A143671 Number of terms in A144688 of length n.

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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).

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A156071 Concatenation chain arising in A156069.

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A214437 Least numbers whose groups of 2,3,...,n digits taken from the left are divisible by 2,3,...,n.

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Mathematica

A305701 Nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

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PARI

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

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A082399 a(1) = 1; thereafter, a(n) is the smallest nonnegative number such that the number Sum_{i=1..n} a(i)*10^(n-i) is divisible by n.

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A271374 Total number of polydivisible numbers in base n.

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Extensions

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

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