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• Sunday, December 05, 2004 11:19 PM [GMT+1=CET], ... The sum of the first n odd cubes is S(n) = Sum((2k-1)^3, k, 1, n) = 2n^4 - n^2 = n^2(2n^2 - 1) If n = 2^m,
Message 1 of 3 , Dec 6, 2004
Sunday, December 05, 2004 11:19 PM [GMT+1=CET],
Michael Gian <work.gian@...> escribió:

> As I have read, for prime numbers, p, 2^p-1 are called Prime-
> Exponent Mersenne numbers.
> These are sub-group of 2^k-1, k positive integers, which are called
> (just) Mersenne numbers.
> When 2^p-1 is prime, it is called a Mersenne prime.
>
> (2^(p-1)*(2^p-1)is a Perfect Number when 2^p-1 is a Mersenne prime.
>
> Does anyone know of a accepted term for a number of the form (2^(p-1)
> *(2^p-1) when 2^p-1 is not prime?
>
> The reason I ask is that all numbers of this form, including Perfect
> Numbers, are the sum of consecutive odd cubes. I am searching for a
> more concise way to talk about them.
>
> Michael

The sum of the first n odd cubes is

S(n) = Sum((2k-1)^3, k, 1, n) = 2n^4 - n^2 = n^2(2n^2 - 1)

If n = 2^m, then

S(2^m) = 2^(2m)(2^(2m+1) - 1) = 2^(p-1)(2^p - 1) = M(p)

Then Mersenne number M(p) = 2^(p-1)(2^p - 1), with odd p, but no neccesary
prime, is the sum of the first 2^((p-1)/2) odd cubes.

Saludos,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa@...
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