When Does the Exponential Function Reach Its Maximum Value?

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Discussion Overview

The discussion revolves around the expression n^( (((n-k)*(k-1)/(2k)) + (k-1))/(n-1) ) and the conditions under which it reaches its maximum value, particularly focusing on the value of k that maximizes this expression. Participants explore theoretical aspects, mathematical reasoning, and potential applications of the function.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants question the definition of n and its constraints, suggesting it is a positive integer less than a limit referred to as "LIM".
  • There is a proposal to define Q_n(k) as the expression in question and to find k_0 such that Q_n(k_0) is maximized for each n, with k_0 appearing to be 36.
  • One participant expresses confusion over their earlier computations, indicating a potential error in their application rather than in the mathematical reasoning itself.
  • Multiple participants analyze the function's behavior at specific values of k, noting that when k=1, k=sqrt(n), and k=n, the function simplifies to k.
  • Some participants reference WolframAlpha to illustrate interesting results and series expansions related to the function.
  • There are discussions about the roots of the function and whether certain statements about them are correct, particularly concerning the symmetry and divisor function related to n.
  • Further exploration includes connections to Euler's Generalized Pentagonal Numbers and the divisor function, with detailed mathematical expressions and results being shared.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the maximum value of k or the implications of their findings. Multiple competing views and interpretations of the function's behavior remain present throughout the discussion.

Contextual Notes

Some statements rely on specific assumptions about the values of n and k, and the discussion includes unresolved mathematical steps and definitions that may affect the conclusions drawn by participants.

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?
 
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What is n?
 
SteamKing said:
What is n?
integers
0<n<LIM
 
^ 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?
 
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?

Yes, you are correct. Thanks for stating it in a better way.
 
JeremyEbert said:
Yes, you are correct. Thanks for stating it in a better way.

Well, I feel real dumb... Nevermind this thread, I had an issue with the application I was using to compute the results. Sorry to all who spent any time on this.
 
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?


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
 
WolframAlpha shows some interesting results at well. The series expansion shows terms involving double factorial numbers.

http://www.wolframalpha.com/widgets/view.jsp?id=daf29fc2857c2b71d6be58dcc6e7ef49

http://www.wolframalpha.com/widgets/view.jsp?id=b8ba4c95900e275211a98d8bd1b0a53c
 
Last edited by a moderator:
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

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
 
Last edited by a moderator:
  • #10
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

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?
 
Last edited by a moderator:
  • #11
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 "
 
Last edited:
  • #13
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?

The divisor symmetry still shows up nicely. For example 36:

(ln(x)/ln(36)) - (((36-x)*(x-1)/(2*x)) + (x-1))/(36-1))

http://dl.dropbox.com/u/13155084/36.png
 
Last edited by a moderator:
  • #14
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
 
  • #15
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 "

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}]
 
  • #17
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

A better way to plot this:

e^(pi i ( ((n-k^2)/(2k)) / ((n-1)/2) ) ) for k=1 to 36, n=36

http://dl.dropbox.com/u/13155084/unit%20circle.gif
 
Last edited by a moderator:

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