

A092175


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.


3



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

1,1


COMMENTS

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.


REFERENCES

Michael Brand was the originator of the problem.


LINKS



FORMULA

When n is even, a(n) = 2*n^(n/2)  n + 1.


EXAMPLE

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 base10 sticky digit labels, but the 199991st tape can't be labeled with 199991 sets of sticky digit labels.


CROSSREFS



KEYWORD

nonn,base


AUTHOR

Ken Bateman (kbateman(AT)erols.com) and Graeme McRae, Apr 01 2004


EXTENSIONS

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


STATUS

approved



