- #1
JeremyEbert
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why is it that the largest value of n^( (((n-k)*(k-1)/(2k)) + (k-1))/(n-1) ) always seems to be when k=36?
integersSteamKing said:What is n?
Dr. Seafood said:^ What's "LIM"? Do you mean n is any positive integer?
I suppose the job to be done here is to find which value of k maximizes the expression in question. Let [itex]Q_n(k)[/itex] be that expression. The problem is to find [itex]k_0[/itex] such that, for each n, [itex]Q_n(k_0) \geq Q_n(k)[/itex] for all k -- and subsequently, to show that apparently [itex]k_0 = 36[/itex]. Is this what you're asking?
JeremyEbert said:Yes, you are correct. Thanks for stating it in a better way.
JeremyEbert said:why is it that the largest value of n^( (((n-k)*(k-1)/(2k)) + (k-1))/(n-1) ) always seems to be when k=36?
JeremyEbert said:Interesting function none the less.
t=(((n-k)*(k-1)/(2k)) + (k-1))/(n-1)
n^(t)
when k=1 then t=0 and n^(t)=k
when k=n^(1/2) then t=0.5 and n^(t)=k
when k=n then t=1 and n^(t)=k
JeremyEbert said:Looking at its deviation from k is very interesting:
k - n^( (((n-k)*(k-1)/(2k)) + (k-1))/(n-1) )
http://dl.dropbox.com/u/13155084/nt-k-4.png
http://dl.dropbox.com/u/13155084/nt-k-9.png
http://dl.dropbox.com/u/13155084/nt-k-16.png
http://dl.dropbox.com/u/13155084/nt-k-25.png
http://dl.dropbox.com/u/13155084/nt-k-36.png
http://dl.dropbox.com/u/13155084/nt-k-49.png
JeremyEbert said:so basically the roots of the function:
log(n,k) - ((((n-k)*(k-1)/(2k)) + (k-1))/(n-1) )
are
k=1
k=n^(1/2)
k=n
Is this a correct statement?
JeremyEbert said:more information
"Another link into Eulers Generalized Pentagonal Numbers and the divisor function d(n):
For our Divisor summatory function we have:
D(n) = SUM(d(n)) :
for k = 0 --> floor [sqrt n]
SUM (d(n)) = SUM ((2*floor[(n - k^2)/k]) + 1)
The notable difference in the equation from the published version is the:
(n - k^2)/k (congruence of squares)
which is derived from the
z = (n - k^2)/2k + i n^(1/2)
forming a parabolic coordinate system.
The function (n - k^2)/2k forms a divisor symmetry centered on the square-root of n.
Example:
k = divisors of n {1,2,3,4,6,9,12,18,36}
n = 36
+17.5, +8.0, +4.5, +2.5, 00.0, -2.5, -4.5, -8.0, -17.5
key results:
sqrt(n) = 0
Sum Terms = 0
Offsetting by -((n-1)/2) = -17.5 and taking the absolute values gives us:
0, 9.5, 13, 15, 17.5, 20, 22, 25.5, 35
Key results:
sqrt(n) = (n-1)/2;
Another way to generate these terms is:
((n-k)*(k-1)/2k) + (k-1)
The key ratio here being the (k-1)/2k function.
reducing this ratio sequence we get:
01/04, 01/03, 03/08, 02/05, 05/12, 03/07, 07/06, 04/09, 09/20, 05/11, 11/24, 06/13, 13/28, 07/15, 15/32, 08/17, 17/36
or
01 01 03 02 05 03 07 04 09 05 11 06 13 07 15 08 17
04 03 08 05 12 07 16 09 20 11 24 13 28 15 32 17 36
Showing a direct connection to Eulers Generalized Pentagonal Numbers and the divisor function d(n)
**
A026741 ( n if n odd, n/2 if n even. ) = xx, 00, 01, 01, 03, 02, 05, 03, 07, 04, 09, 05, 11, 06, 13, 07, 15, 08, 17
A022998 ( If n is odd then n else 2*n. ) = 00, 01, 04, 03, 08, 05, 12, 07, 16, 09, 20, 11, 24, 13, 28, 15, 32, 17, 36
A026741 = Partial sums give Generalized Pentagonal Numbers A001318 = 00, 01, 02, 05, 07, 12, 15, 22, 26, 35, 40, 51, 57
A022998 = Partial sums give Generalized Octagonal Numbers A001082 = 00, 01, 05, 08, 16, 21, 33, 40, 56, 65, 85, 96, 120 "
JeremyEbert said:the contour plot shows the divisor function very nicely:
(n-k^2)/2k mod .5
http://www.wolframalpha.com/input/?i=ContourPlot[Mod[(-k^2+++n)/(2+k),+0.5],+{k,+-2,+2},+{n,+-4,+4}]
JeremyEbert said:for the function
f(n,k) = ( (ln(k)/ln(n)) - ((((n-k)*(k-1)/(2k)) + (k-1))/(n-1) )
the
local minimum = (((n-1)/2)-sqrt(((n-1)/2)^2-(n*ln^2(sqrt(n)))))/log(sqrt(n))
local maximum = (((n-1)/2)+sqrt(((n-1)/2)^2-(n*ln^2(sqrt(n)))))/log(sqrt(n))
ex: n=49
49 minimum = (24-sqrt(576-49 log^2(7)))/(log(7))
49 maximum = (24+sqrt(576-49 log^2(7)))/(log(7))
and min * max = n
An exponential function is a mathematical function in the form of f(x) = ab^x, where a and b are constants and x is a variable. It is a type of non-linear function in which the input variable appears in the exponent. The graph of an exponential function is a curve that increases or decreases rapidly.
The main difference between an exponential function and a linear function is that the variable in an exponential function appears in the exponent, while in a linear function it appears in the base. This results in a curved graph for an exponential function and a straight line for a linear function.
An exponential function is used to model situations where the rate of change is proportional to the current value. It is commonly used in finance, population growth, radioactive decay, and other natural phenomena. It is also used in computer science and engineering to analyze algorithms and growth of data.
To solve an exponential function, you can use logarithms, which are the inverse of exponential functions. By taking the logarithm of both sides of the equation, you can isolate the variable and solve for it. Another method is to use a graphing calculator or software to find the intersection of the exponential function and a straight line.
Some of the properties of exponential functions include:
1. The domain is all real numbers.
2. The range is positive real numbers if the base is greater than 1, or negative real numbers if the base is between 0 and 1.
3. The function is continuous and differentiable for all real numbers.
4. The graph is always increasing or decreasing, depending on the value of the base.
5. The x-intercept is always 0, and the y-intercept is 1.
6. The function has a horizontal asymptote at y = 0.