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- W1549447656 abstract "This paper is based on an article in Mathematical Spectru.m, VoL 29, No 1. It concerns what happens when an operation applied to an n-digit integer results in an n digit integer. Since the number of ndigit integers is finite a repetition must occur after applying the operation a finite number of times. It was assumed in the above article that this would lead to a periodic sequence which is not always true because the process may lead to an invariant. The second problem with the initial article is that, say, 7 is considered as 07 or 007 as the case may be in order make its reverse to be 70 or 700. However, the reverse of 7 is 7. In order not to loose the beauty of these sequences the author has introduced stringent definitions to prevent the sequences from collapse when the reversal process is carried out. Four different operations on n-digit integers is considered. The Smarandache n-digit periodic sequence. Definition: Let Nlc be an integer of at most n digits and let ~ be its reverse. N1,,' is defined through N~ = Rk ·1Olt-l-[logloN1 ] . The element Nlo+l of the sequence through Nlo+1= I Nlc-N1,,' I where the sequence is initiated by an arbitrary n-digit integer Nl in the domain lOn~l I be a fixed integer and No and aibitrary positive integer. Nlo+l is derived from Nlc by multiplying each digit x of Nlc by c retaining only the last digit of the product cx to become the corresponding digit of Nlo+1• The Smarandache Mixed Composition Periodic Sequence: Definition. Let No be a two-digit integer al·lO+ao. If aI+ao<lO then bl= al+ao otherwise bl= aI+ao+1. bo=lal-aol. We define Nl=bI·IO+bo. Nlc+l is derived from Nlc in the same way Starting points for loops (periodic sequences), loop length and the number of loops of each kind has been calculated and displayed in tabular form in all four cases. The occurrence of invariants has also been included." @default.
- W1549447656 created "2016-06-24" @default.
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- W1549447656 date "1997-08-01" @default.
- W1549447656 modified "2023-09-23" @default.
- W1549447656 title "On Smarandache's periodic sequences" @default.
- W1549447656 doi "https://doi.org/10.5281/zenodo.9468" @default.
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