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Thread: What is the name of this sequence/"function"?

  1. #1
    Newbie Feryll's Avatar
    Nov 2009

    What is the name of this sequence/"function"?

    This is the number sequence a(x) whose outputs are determined by the greatest integer divisors n of any factorizations of x with the regulation that if a factor is repeated in the factorization, the factorization is unacceptable. Repeat elements welcome.

    EX: a(56). The valid factorizations are 1*56, 2*28, 4*14, 7*8, 7*2*4. The illegal facotrs, 7*2*2*2 and 2*2*14, would not be allowed due to the repeated 2s. The output elements thus are the highest elements in the valid factorizations: 7,8,14,28,56. Not applicable in this example is that if you end up with two factorizations, unique to each other beyond permutation of factors, such as 2*3*7 and 7*6 of 42, you must include the greatest factor twice in the elemental output (eg a(84)={7, 7, 12, 14, 21, 28, 42, 84}).

    This sequence is consequential to the series below:
    $\displaystyle \prod_{n=1}^T (1+n^{-1})$

    Which is:


    This is just T+1, clearly, but if you manually multiply out the terms, you'll get, for example for t=4, this:


    As you see, the expansion of the series for term T is equivalent to adding all of the reciprocals of the elements of the function on a graph of our curious number sequence graph below or on the line y=T, and then multiplying the result by 2. This is purely because of how we defined the sequence/algorithm in the first place.

    What I'm asking, plain and simple, is, "what is the name of this sequence/where is there study into this?" or "is this a trivial problem/sequence?" since I'm just getting into number theory myself.

    Here are a couple of things I've been able to notice about this, but they're almost trivial:

    1. No elements exists above the line of y=x.
    2. No elements exist below the line of y=infa(x) where infa(x) is the inverse of the factorial (sorry, I don't know my terminology).
    3. The addition of the reciprocals of the elements' x-values on any y-level with y being any natural number above 1 is 1/2.
    4. The elements' x-values, with sole elements (thus existing on the line y=x), are all either primes or square primes, and it includes each and every prime and square prime.
    5. I have a couple of results for calculating the elements existing on a line y=x/k for any natural number k with relation to the values existing on y=x (ie the primes and the squares of primes), although it is only a primitive process which produces a subset of the elements.

    I would create and display a graph, but I can't really capture what I'm trying to show here on such a surface. I will try to if someone asks for one.

    I apologize in advance for any terminology or drunken logic errors, as I don't communicate with other math people very often, and my study in these areas are fully solitary.
    Last edited by Feryll; Dec 3rd 2011 at 05:52 PM.
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