about a new smarandache type sequence

8/12/2019 About a new Smarandache type sequence

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About a new Smarandache type sequenceCsaba Biro, University oJSzeged

In this paper we will discuss about a problem that I asked about 8 years ago,when I was interested mainly in computer science. The computers can operate with256 characters and all of them h s an ASCn code which is an integer from 0 to 255. Ifyou press AL T key and you type a number, the character of the number will appear.But if you type a number that is greater than 255, the computer will calculate theremainder after division by 256, and the corresponding character will appear. Canyou show each character by pressing the same number key k-times? - asked I

It is quite simple to solve this problem, and the answer is no. Before provingthis we generalize the problem to t-size ASCn code-tables, the codes are from 0 to II

We shall use the following notations: N is the set of the positive integers,NO=Nu{O}, Z is the set of the integers and Zt={O l t-l}.

Now let us see the generalized problem. e f i n e f N ~ N as

where

I {X EZ t :alO X modt) for some k ENo and a E{O I ... ,9}}1=0

Our first question was j{256), and the generalized problem is to calculate j{in generality.

It is clear thatj{t =t if ts10, a n d j { t ~ 1 0 if t>1O Now let us examine somespecial cases.Let t=2r5s, r,sENo but at least one of them is not zero. Denote by w the

maximum ofr and s. If ~ w , then t/lok, because lok=2k5k. Sok .. IaLlOi EaLHY (mod t ,;0 ;0

thus

So I H [ I ~ l O w , moreover I H t l ~ 9 w 1 because if a=O, then the value of k isinsignificant. 84


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We got a sufficient condition for j{t)9w+1. t is satisfied if r?6 ors?2 or r=2,3,4,5 and s=1. If r=O,1 and s


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Now we still have the question: for which numbers fit =t? Are there finite orinfinite many t with the property above? Is there a better faster) algorithm tocalculate fit ? Is there an explicit formula? Can anyone answer?

BffiLIOGRAPHY[ ] GYARMATI Edit-TURAN Pcil: SzamelmeletNemzeti Tankonyvkiad6, Budapest, 1995.[2] Fritz REINHARD-Heinrich SOEDER: dtv-Atlas zur Mathematik

Deutscher Taschenbuch Verlag GmbH Co. Kg, MUnchen, 1974. and 1977.

Address:J zsejAltila TudomimyegyetemBolyai Inte=etAradi vertanuk tere 16720 SzegedHUNGARY

E-mail:[email protected]

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about a new smarandache type sequence

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