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Proof that ∏(√n) increments when n is a centered polygonal number with a prime index.

  1. Oct 15, 2012 #1
    Is there a proof that ∏(√n) increments only when n is a centered polygonal number with a prime index?
    ∏(n) is the prime counting function
    n=p^2-p+1 for a prime p
    3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333....
    http://oeis.org/A119959
     
  2. jcsd
  3. Oct 16, 2012 #2
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    The statement in your question is not very precise. Obviously, ∏(√n) increments exactly when n=p2, where p is a prime, and for no other n. But I suppose that is not what you are looking for, and I am curious to know what your centered polygonal numbers have to do with this.
     
  4. Oct 16, 2012 #3
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    Norwegian,
    Sorry for the extreme lack of precision. I definitely forgot to include a key part while typing this up.
    Obviously, ∏(√n) increments exactly when n=p2. However ∏( Floor(√n + 0.5) ) increments only when n is a centered polygonal number with a prime index. It seems like a proof of this should be pretty straight forward.
     
  5. Oct 16, 2012 #4
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    Hi Jeremy, if we instead look at ∏(1/2 + √n), this function jumps exactly when n=(p-1/2)2=p2-p+1/4. If we require n to be an integer, the jumps will indeed happen when n=p2-p+1.
     
  6. Oct 16, 2012 #5
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    Ah yes, the floor function is a bit redundant. Thanks for that.
    This obviously relates then to the square root of a oblong (pronic) number + 1/4 having a half-integer value. The name "oblong" indicating they are analogous to polygonal numbers.
     
  7. Oct 16, 2012 #6
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    I'm not sure what you mean by "analogous to polygonal numbers." Oblong numbers map 1:1 to triangular numbers (= 2 times a triangular number) and triangular numbers are one kind of polygonal number, but Polygonal numbers in general?

    It's true that the polygonal numbers can be constructed from the counting numbers + multiples of triangular numbers, but that involves first differences between polygonal numbers of equal index (e.g. the 5th square is 25 and 25 + 10, the fourth triangular number, = 35, the 5th Pentagonal number), not polygonal numbers themselves.

    - AC
     
  8. Oct 16, 2012 #7
    Re: Proof that ∏(√n) increments when n is a centered polygonal number with a prime in

    A pronic + 1 is a centered polygonal number. ∏(1/2 + √n) gives no information about ∏ itseft, it is trivial.
     
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