![]() ![]() Also, for some numbers we do not know if there is a cycle. Thus, TITO(291) is the reverse of 3*79, which is equal to 732. Take for example 291 which is a product of primes 3 and 97. If n is not a palindrome and the reverse of n has only palindromic factors, then the trajectory of n is a two-cycle. For numbers n that have only palindromic factors, TITO(n) is equal to the reverse of n. There are numbers that generate two-cycles. There are numbers that have one-cycle destinies, but are not the fixed points of TITO operation themselves. ![]() There are other cases too: for example 26 is a fixed point, but is neither prime nor palindromic. For example, a palindrome that is a product of palindromic primes is a fixed point of the TITO operation. There are numbers other than prime that have one-cycle destinies. That means all prime numbers have different destinies of the same type: they end with a one-cycle. ![]() It is easy to see that prime numbers are among the fixed points of the TITO operation. Then we reverse the result: TITO(68) = 482. We reverse them and multiply: 2*2*71 = 284. For example, to calculate TITO(68), we first find prime factors of 68, which are 2, 2 and 17. By definition, to calculate TITO( n) you need to reverse the prime factors of n, multiply them back together (with multiplicities) and reverse the result. TITO is an abbreviation of “Turning Inside, Turning Outside”. The next interesting example is the TITO operation. Even when we know what kind of life the number is living the destinies are not always clear.Ĭase 5. ![]() The sequence of new destinies starts 1, 6, 28, 220, … and we do not know what the next number is because for 276 we do not know the behavior of its trajectory. Thus all the numbers whose trajectories are finite have the same destinies. For our example of sums of proper divisors all finite sequences end with 0. This situation makes the definition of destinies more complicated, but it is appropriate to say that finite sequences have the same destinies if they end with the same number. So, let us say the trajectory of 15 is finite, and ends with 1, 0. The sum of proper divisors of zero is not-defined or is equal to infinity, whichever you prefer. Let’s look at the trajectory of 15: it is 15, 9, 4, 3, 1, 0. For the next example, let f(n) denote the sum of proper divisors. For this operation we have two types: one-cycles for palindromic integers and two-cycles for non-palindromic integers.Ĭase 4. In this case, instead of studying destinies, it might be more interesting to study types of destinies. The first appearance of a new destiny is sequence A131058 - a list of numbers n whose reverse is not less than n. If a number is not a palindrome, then its trajectory is a two-cycle consisting of a number and its reverse. If a number is a palindrome, then its trajectory is a one-cycle consisting of that number. Therefore, with respect to this function all integers have the same destiny.Ĭase 3. Then wherever we start, the tail of the trajectory with respect to the function f(n) is a sequence of consecutive prime numbers. Let the function f(n) be the next prime after n. For the SOD operation, the sequence of the first occurrences of new destinies is finite and is equal to: 1, 2, 3, 4, 5, 6, 7, 8, 9.Ĭase 2. This is the sequence of numbers c(n) such that c(n) is the smallest number with its destiny. Given an operation, we can build another sequence that is called “the first occurrence of a new destiny”. It follows that all the natural numbers have 9 different destinies with respect to SOD, which only depend on the remainder of the number modulo 9. In the above example of SOD, any trajectory of a positive integer ends with a one-digit non-zero number repeating many times. In particular, all numbers in the same trajectory a(n) have the same destiny. Then the numbers a(0) and b(0) on which these trajectories are build have the same destiny if there exist N and M, such that for any j, a(N+j) = b(M+j). Suppose a(n) and b(n) are two trajectories. Two numbers have the same destiny with respect to SOD if the tails of their trajectories coincide. Then the trajectory of a number k is the sequence a(n), such that a(0) = k and a(n+1) = SOD(a(n)). Suppose our function is the sum of digits of a number, denoted as SOD. It would be harmonious to assume that he suggested this term.Ĭase 1. I know the term “destiny” from John Conway, the creator of the Game of Life. Do you know that numbers have destinies? Well, to have a destiny, a number needs to have a life, or in mathematical terms, destinies are defined with respect to an operation or a function. ![]()
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