Search: a158242 -id:a158242
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.
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LINKS
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Albert Franck, Puzzles, see item 7.
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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STATUS
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approved
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A214437
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Least numbers whose groups of 2,3,...,n digits taken from the left are divisible by 2,3,...,n.
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+10
4
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1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800
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OFFSET
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1,2
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COMMENTS
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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 %.
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
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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STATUS
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approved
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A305714
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Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.
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+10
4
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1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 1
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}
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CROSSREFS
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Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305715.
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KEYWORD
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nonn,full,base,fini
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AUTHOR
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STATUS
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approved
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A305715
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Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.
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+10
3
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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polyQ[q_]:=And@@Table[Divisible[FromDigits[Take[q, k]], k], {k, Length[q]}];
Flatten[Table[Select[Permutations[Range[n]], polyQ], {n, 8}]]
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CROSSREFS
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Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305714 (row lengths).
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KEYWORD
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base,fini,tabf,full,nonn
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AUTHOR
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STATUS
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approved
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A158240
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Smallest number made up of n consecutive digits such that every k-digit substring (k <= n) taken from the left is divisible by k (k=1..n).
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+10
1
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OFFSET
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1,2
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LINKS
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EXAMPLE
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123654, for instance, is in the sequence because 1|1, 2|12, 3|123, 4|1236, 5|12365, 6|123654.
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CROSSREFS
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KEYWORD
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nonn,base,fini,full
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AUTHOR
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EXTENSIONS
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a(4), a(5), a(7), a(9) corrected by Ray Chandler, Mar 21 2009
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STATUS
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approved
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A331475
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a(n) is the smallest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).
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+10
1
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OFFSET
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1,2
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COMMENTS
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a(n) = A147636(n) for n=1, 2, 3, 9 and 10.
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LINKS
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EXAMPLE
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a(3) = 3012 because 2, 12, 012, 3012 are divisible by 1, 2, 3, 4 and it is the least such number with distinct digits 0 to 3.
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MATHEMATICA
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ok[n_] := AllTrue[Range@ IntegerLength@ n, Mod[ Mod[n, 10^#], #] == 0 &]; a[n_] := SelectFirst[ FromDigits /@ Permutations[Range[0, n-1]], # >= 10^(n-1) - 1 && ok[#] &]; Array[a, 10] (* Giovanni Resta, May 04 2020 *)
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CROSSREFS
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KEYWORD
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base,fini,full,nonn
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AUTHOR
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STATUS
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approved
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A334537
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a(n) is the largest n-digit number using each digit 0 to n-1 once, such that the numbers formed by its last k digits are divisible by k, (k = 1..n).
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+10
0
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OFFSET
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1,2
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LINKS
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EXAMPLE
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a(4) = 43120 because 0, 20, 120, 3120 and 43120 are divisible by 1, 2, 3, 4 and 5, and it is the largest such number with distinct digits 0 to 4.
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CROSSREFS
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KEYWORD
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base,fini,full,nonn
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AUTHOR
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STATUS
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approved
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