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A092175
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Define d(n,k) to be the number of '1' digits required to write out all the integers from 1 through k in base n. E.g., d(10,9) = 1 (just '1'), d(10,10) = 2 ('1' and '10'), d(10,11) = 4 ('1', '10' and '11'). Then a(n) is the first k >= 1 such that d(n,k) > k.
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3
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2, 3, 13, 29, 182, 427, 3931, 8185, 102781, 199991, 3179143, 5971957, 114818731, 210826995, 4754446861, 8589934577, 222195898594, 396718580719, 11575488191148, 20479999999981, 665306762187614, 1168636602822635, 41826814261329723, 73040694872113129
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OFFSET
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1,1
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COMMENTS
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The number of video tapes you can label sequentially starting with "1" using the n different number stickers that come in the box, working in base n.
Adapted from puzzle described in the Ponder This web page.
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REFERENCES
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Michael Brand was the originator of the problem.
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LINKS
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FORMULA
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When n is even, a(n) = 2*n^(n/2) - n + 1.
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EXAMPLE
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John Fletcher gives the following treatment of the case of odd B at the 'solutions' link: a(10)=199991 because you can label 199990 tapes using 199990 sets of base-10 sticky digit labels, but the 199991st tape can't be labeled with 199991 sets of sticky digit labels.
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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Ken Bateman (kbateman(AT)erols.com) and Graeme McRae, Apr 01 2004
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EXTENSIONS
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Edited by Robert G. Wilson v, based on comments from Don Coppersmith and John Fletcher, May 11 2004
a(13) corrected and a(23) onwards added by Gregory Marton, Jul 29 2023
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STATUS
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approved
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