st two members. If it has only one member it will be referred to as invariant. There is one more snag to overcome. In the Smarandache sequences OS is considered as a two-digit integer. The consequence of this is that 00056 is considered as as a five digit integer while 056 is considered as a three-digit integer. We will abolish this ambiguity, 05 is a one-digit integer and 00200 is a three-digit integer. 6050:T5c4,With these two remarks in mind let's look at these sequences. There are in all four different ones reported in the above mentioned article in Mathematical Spectrum. The study of the first one will be carried out in much detail in view of the above remarks. IV.2 The Two-Digit Smarandacbe Periodic Sequence It has been assumed that the definition given below leads to a repetition according to Dirichlet's box principle (or the statement made above). However, as we will see, this definition leads to a collapse of the sequence. Preliminary dermition. Let Nit be an integer of at most two digits and let N k ' be its digital reverse. We define the element Nk+1 of the sequence through where the sequence is initiated by an arbitrary two digit integer N 1 â€˘ Let's write NI in the form N 1=10a+b where a and b are digits. We then have N 2= 110a+b-l0b-a 1=9Âˇ1 a-b 1 The 1a-b 1can only assume 10 different values 0,1,2, ... ,9. This means that N3 is generated from only 10 different values of N2 . Let's first find out which two digit integers result in 1a-b 1= 0,1,2, .. and 9 respectively. Ia-bl 0 1 2 3 <4 5 6 7 8 9 11 10 13 1<4 15 16 17 19 19 Corresponding two digit integers 22 33 ÂŤ 55 60 77 88 12 21 23 32 34 43 <45 20 2<4 31 35 <42 <46 53 25 30 36 <41 <47 52 58 26 37 .w <48 51 59 62 27 38 <49 50 61 72 83 28 39 60 71 82 93 92 29 70 81 80 91 90 61 99 5-4 57 56 63 69 73 9<4 8<4 6<4 65 68 67 75 7<4 95 85 76 79 96 78 86 87 97 89 9851:T405,Conclusion: The iteration always produces a loop of length 5 which starts on the second or the third term of the sequence. The period is 9,81,63

Computer Analysis of Number Sequences