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Search: id:A079505
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| A079505 |
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The last number for which a determinant of base n numbers is nonzero. |
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+0
1
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| 14, 25, 61, 121, 211, 337, 505, 721, 991, 1321, 1717, 2185, 2731, 3361, 4081, 4897, 5815, 6841, 7981, 9241, 10627, 12145, 13801, 15601, 17551, 19657, 21925, 24361, 26971
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OFFSET
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2,1
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COMMENT
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Suppose the number k written in base b requires N digits. We build A_k, a square N X N matrix with the digits of k, k-1,...,k-N+1 in base b. The number Det[A_k] is 0 for k greater than b^3-b+1 (except if b=2).
|Det[A_k]| is at most (b-1)^2. The last nonzero value is 1-b, which occurs for k = b^3-b+1 (cf. A061600) except for b=2, though I did not prove it.
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FORMULA
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a(n)=n^3-n+1 (except for n=2, a(2)=14)
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EXAMPLE
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a(3)=25 because the determinant sequence in base 3 is 1, 2, 2, -1, -1, 4, -2, -2, -2, 2, 0, 1, -1, 0, 1, -1, 0, -4, 4, 0, 2, -2, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0.... and Det[A_k]=0 for k > 25.
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MAPLE
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Table[ls = {}; Do[nt = Length[IntegerDigits[k, b]]; Ak = Table[IntegerDigits[k - i, b, nt], {i, 0, nt - 1}]; AppendTo[ls, Det[Ak]], {k, 1, b^4}]; Position[ls, _?(#!=0&)][[ -1, 1]], {b, 2, 10}]
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CROSSREFS
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Cf. A061600.
Sequence in context: A164401 A164387 A164398 this_sequence A039604 A030786 A094163
Adjacent sequences: A079502 A079503 A079504 this_sequence A079506 A079507 A079508
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KEYWORD
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base,nonn
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AUTHOR
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Carlos Alves (cjsalves(AT)gmail.com), Jan 21 2003
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EXTENSIONS
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Edited by T. D. Noe (noe(AT)sspectra.com), Jun 24 2009
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