The MRB constant is defined at
http://mathworld.wolfram.com/MRBConstant.html .
First enter
mRB = 0.18785964246206712024851793405427323005590309490013878617200468\
4089477231564660213703296654433107496903;
lmRB = Table[
FractionalPart[N[1/mRB*(5060936308 + 78389363/24*n), 40]], {n, -10,
10}];
Then enter
Table[lmRB[[n]] - lmRB[[n - 2]], {n, 3, 20}]
And get
{-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, 0.99999999954747953811735414295, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10, \
-4.5252046188264585705*10^-10, -4.5252046188264585705*10^-10}
(Almost all elements are close to 0.)
But if you try to first enter
lE = Table[
FractionalPart[N[E*(5060936308 + 78389363/24*n), 40]], {n, -10,
10}];
and
Table[lE[[n]] - lE[[n - 2]], {n, 3, 20}]
You get
{-0.25105168103432683246864866987, 0.74894831896567316753135133013, \
0.74894831896567316753135133013, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, 0.74894831896567316753135133013, \
0.74894831896567316753135133013, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, -0.25105168103432683246864866987, \
-0.25105168103432683246864866987, 0.74894831896567316753135133013}
(elements that are far from 0)..
Even if you miss MRB by a very little bit by entering
mRBnear = mRB - 10^-10;
lmRBnear =
t = Table[
FractionalPart[
N[1/mRBnear*(5060936308 + 78389363/24*n), 40]], {n, -10, 10}];
and then
Table[lmRBnear[[n]] - lmRBnear[[n - 2]], {n, 3, 20}]
you get
{0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057, \
0.01851010558344280885973117057, 0.01851010558344280885973117057}
(elements that are far from 0)!
I would like to explain why! but if you can beat me to it, please go ahead!!
The answer isn't that all the elements of lmRB are already close to 0, though that route is a strong tempation.
When you then enter
lmRB
you get
{0.50000000249060737870114483554,
2.26434714775982190702*10^-9, 0.50000000203808691681849897849,
1.81182668587717604997*10^-9, 0.50000000158556645493585312145,
1.35930622399453019292*10^-9, 0.50000000113304599305320726440,
9.0678576211188433588*10^-10, 0.50000000068052553117056140735,
4.5426530022923847883*10^-10, 0.50000000022800506928791555031,
1.74483834659262178*10^-12, 0.49999999977548460740526969326, \
0.99999999954922437646394676474, 0.49999999932296414552262383621, \
0.99999999909670391458130090769, 0.49999999887044368363997797917, \
0.99999999864418345269865505064, 0.49999999841792322175733212212, \
0.99999999819166299081600919360, 0.49999999796540275987468626507}
(elements that are close to 0, but also 1, and 1/2). As for a good reason, I am working on it. I will even accept your guesses!
EDIT:
With lmRB[[n]] - lmRB[[n - 2]] being close to 0, for the most part, It makes some sense that if you enter
Table[lmRB[[n]] - lmRB[[n - 1]], {n, 3, 20}]
you get
{0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, -0.50000000022626023094132292852, \
0.49999999977373976905867707148, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148, \
-0.50000000022626023094132292852, 0.49999999977373976905867707148}
(All elements that are close to +/- 1/2).
For brevity, I limited the domain of the tables to [-10,10], but I get very similar results all the way from -1000 to 1000.
Try
Table[FractionalPart[N[1/mRB*(5060936308 + 78389363/24*n), 40]], {n, -1000, 1000}]