From these sets we see that  10121872616 / 53879973811 - MRB constant ~= 0.000000000000000000001.

Try

for Digits from 2 to 23 do m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); print(10121872616/53879973811-m-0.1e-20) end do

or simply View 565_10ton22.mw on MapleNet or Download 565_10ton22.mw
View file details  .

A fourth large independent set of approximations of the form sin((a+b*n)*pi/m) involving the MRB constant that produce multiple repeating numbers at periods of one, is sin((22370113607+22291724244*n)*Pi/m). A fifth is sin((5437555970+5359166607*n)*Pi/m). A sixth is sin((22746733269+78389363*n)*Pi/m) .

Combining the first set with these we have a super set of   sin((5060936308+376619662*n)*Pi/m) .

for n from -1000 to 1000 do u := evalf(sin((5060936308+376619662*n)*Pi/m)); if abs(-1-u) < 10^(-16) then print(n, "   ", u) end if end do

gives

-7, " ", -0.99999999999999991050140000351636098218261268432391282800431707

-6, " ", -0.99999999999999993306721915923629144965641850430406591650488712

-5, " ", -0.99999999999999995236072039788282566260363456406271141231290650

-4, " ", -0.99999999999999996838190371945596355788979107729902762706123721

-3, " ", -0.99999999999999998113076912395570508308848280811176444216094387

-2, " ", -0.99999999999999999060731661138205019648136907099924348035695069

-1, " ", -0.99999999999999999681154618173499886705817373085935824224385762

0, " ", -0.99999999999999999974345783501455107451668520298957420774136611

1, " ", -0.99999999999999999940305157122070680926275645308692890252955689

2, " ", -0.99999999999999999579032739035346607241030499724803192944401587

3, " ", -0.99999999999999998890528529241282887578131290196906496483074639

4, " ", -0.99999999999999997874792527739879524190582678414578171986086361

5, " ", -0.99999999999999996531824734531136520402195781107350786680503863

6, " ", -0.99999999999999994861625149615053880607588170044714093026815590

7, " ", -0.99999999999999992864193772991631610272183872036115014338300665

8, " ", -0.99999999999999990539530604660869715932213368930957626896451810  .

There might be an infinite number of large independent sets of approximations of the form sin((a+b*n)*pi/m) involving the MRB constant that produce multiple repeating numbers at periods of one.

I found a third one; it also gives consistent results when the coefficients are divided by two and three like the previous two sets did. It is 8737225085429350+1753598*n. a=873722508542935 and b=1753598.

View 565_third_set.mw on MapleNet or Download 565_third_set.mw
View file details

 Further numeric investigation shows the coefficients 5060936308, 78389363 and 996509389 can be replaced by 1/2 and 1/3 of their value and still produce remarkable results.

View 565_MRB_various_fractions.mw on MapleNet or Download 565_MRB_various_fractions.mw
View file details

To sum up what I've noticed so far.

Concerning the MRB constant,

m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)):

From the first  and fourth posts I have come up the following:

restart; Digits := 22; m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); for n from -10 to 10 do p := n; printf("%a %a\n", p, evalf(sin(Pi*(5060936308+78389363*floor(n))/m))) end do

That gives the first set of near-integer equations represented by:

-10    -.9999999999999853247439
-9    -.9999999999999881012605
-8   -.9999999999999905882557
-7    -.9999999999999927828150
-6    -.9999999999999946864810
-5    -.9999999999999963001138
-4    -.9999999999999976218224
-3    -.9999999999999986526377
-2    -.9999999999999993929081
-1    -.9999999999999998417661
0    -.9999999999999999997307
1    -.9999999999999998668019
2    -.9999999999999994426460
3    -.9999999999999987277599
4    -.9999999999999977219805
5    -.9999999999999964244621
6    -.9999999999999948367254
7    -.9999999999999929580953
8    -.9999999999999907872146
9    -.9999999999999883266272
10    -.9999999999999855751465
 

Form the fifth post:

restart; Digits := 22; m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); for n from -10 to 10 do p := 2*n-1; printf("%a %a\n", p, evalf(-sin(996509389*p*Pi/m))) end do:

Wich gives the second set:

-21    -.9999999999999972039693
-19    .9999999999999977153722
-17    -.9999999999999981691216
-15    .9999999999999985780284
-13   -.9999999999999989307085
-11    .9999999999999992332081
-9    -.9999999999999994855272
-7    .9999999999999996901603
-5    -.9999999999999998417661
-3    .9999999999999999429076
-1    -.9999999999999999936564
1    .9999999999999999936564
3    -.9999999999999999429076
5    .9999999999999998417661
7    -.9999999999999996901603
9    .9999999999999994855272
11    -.9999999999999992332081
13    .9999999999999989307085
15    -.9999999999999985780284
17    .9999999999999981691216
19    -.9999999999999977153722

My question to any one who knows is, aren't these two sets independent?

I mean something like, neither one is true just because the other is, right?

The first involved

sin(Pi*(5060936308+78389363*floor(n))/m)

The second involved

sin(996509389*p*Pi/m)

Am I correct in figuring that the periods should be totally unrelated?

And has anyone here had any analysis that explains why these two sets exists? And do you think they can help me find a closed form for the MRB constant?

Let p be any odd number, as an example, say abs(P)<100, then -sin(996509389*p*Pi/m)  = "+/-1. " .

To see this try

restart; Digits := 62: m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)):for n from -100 to 100 do p := 2*n-1; printf("%a %a\n", p, evalf(-sin(996509389*p*Pi/m))) end do:

And with half the accuracy,

let p be any even number, as an example, say abs(P)<1000, then -sin(996509389*p*Pi/m) =0.

To see this try

restart; Digits := 62: m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)):for n from -100 to 100 do p := 2*n; printf("%a %a\n", p, evalf(-sin(996509389*p*Pi/m))) end do:

.

restart; for Digits from 2 to 20 do m := evalf(sum((-1)^i*(i^(1/i)-1), i = 1 .. infinity)); print(evalf(1+sin(5060936308*Pi/m), Digits+5)) end do:restart:

Notice how suddenly the result drops to 10^(-21).

See what happens when you use a domain of complex numbers for n:

View 565_near0_complex.mw on MapleNet or Download 565_near0_complex.mw
View file details

I was so excited I forgot to attach the worksheet. Here it is.

View 565_near0.mw on MapleNet or Download 565_near0.mw
View file details

My conjecture is false.

See http://marvinrayburns.com/latest.html.

For x>0and f(x) = (-1)^x * x^(1/x) , the next document,
Download part_2_sum_vs_int.mw
shows the difference of  and the ratio of abs(int(f(x), x = 1 .. 2*N))-1/2  and sum(f(n),n=1..2*N) as [even] integer N -> infinity. The difference seems to go to 0 and the ratio seems to go to 1. However, it would be nice to have a proof of some sort to show that they both converge upon the same number. I have given a simple geometric example of the summing of f in the following: http://marvinrayburns.com/what_is_mrb.mht. Perhaps a similar example of the integrating of f could help make the proof -- that is if it is true.

The MRB constant=0.187859...

Download part_2_sum_vs_int.mw

Joe Riel, I used your program to make another one that shows the bifurcation going to (e^e, 1/e) .

restart; with(plots): f := proc (x, n) local y, z; z := 1./x; y := z; to n do y := z^y end do end proc:

plt1 := animate(plot, [rcurry(f, 100*a), 10.0 .. 15.2], a = 1 .. 8):

plt2 := animate(plot, [rcurry(f, 100*a+1), 10.0 .. 15.2], a = 1 .. 8):

plt3 := plot([1/exp(1)], 10.0 .. 15.2, gridlines = true, linestyle = dash, color = blue):

print(display(plt1, plt2, plt3, view = [9.9 .. 15.3, .35 .. .40])):

Download 565_tetrations4.mw
View file details

My program takes a couple of minutes to compute.

I'm refering to the HTML version. If you go to Recent Files and click multi powers 2.mw or test.mw, only a small part of the worksheet is shown.