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Search: id:A134640
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| A134640 |
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Permutational numbers (numbers with k different digits in k-positional system). |
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+0
18
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| 0, 1, 2, 5, 7, 11, 15, 19, 21, 27, 30, 39, 45, 54, 57, 75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 194, 198, 214, 222, 238, 242, 294, 298, 334, 346, 358, 366, 414, 422, 434, 446, 482, 486, 538, 542, 558, 566, 582, 586
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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a(1) is the 1-positional system 1!=1 numbers
a(2) to a(3) are two=2! 2-positional system numbers
a(4) to a(9) are six=3! 3-positional system numbers
a(10) to a(33) are 24=4! 4-positional system numbers
a(34) to a(153) are 120=5! 5-positional system numbers
...
There are a(!k)-a(Sum[m!,1,k])=a(A003422)-a(A007489) k-positional system k! numbers
The name permutational numbers arises because each permutation of k elements is isomorphic with one and only one of member of this sequence and conversely each number in this sequence is isomorphic with one and only one permutation of k elelmnts or its equivalent permutation matrix.
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EXAMPLE
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We build permutational numbers:
a(1)=0 in unitary positional system we have only one digit 0
a(2)=1 because in binary positional system smaller number with two different digits is 01 = 1
a(3)=2 because in binary positional system bigger number with two different digits is 10 = 2 (binary system is over)
a(4)=5 because smallest number in ternary system with 3 different digits is 012=5
a(5)=7 second number in ternary system with 3 different digits is 021=7
a(6)=11 third number in ternary system with 3 different digits is 102=11
a(7)=15 120=15
etc.
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MATHEMATICA
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a = {}; b = {}; Do[AppendTo[b, n]; w = Permutations[b]; Do[j = FromDigits[w[[m]], n + 1]; AppendTo[a, j], {m, 1, Length[w]}], {n, 0, 5}]; a (*Artur Jasinski*)
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CROSSREFS
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Cf. A003422, A007489.
Sequence in context: A133436 A001225 A157001 this_sequence A032616 A006066 A084935
Adjacent sequences: A134637 A134638 A134639 this_sequence A134641 A134642 A134643
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KEYWORD
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nonn
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AUTHOR
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Artur Jasinski (grafix(AT)csl.pl), Nov 05 2007, Nov 07 2007, Nov 08 2007
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EXTENSIONS
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Corrected indices in examples. Replaced dashes in comments by the word "to" - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 26 2009
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