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## Observation: A Prime / Mersenne / (Ramanujan) Triangular Number Convolution

for...

p'_n = {1 Union Prime Numbers}
M_n = n-th Mersenne Number (2^n - 1)
T_n = n-th Triangular Number (n^2 + n)/2

x = {0,1,2,3,13} --> F_(0, 1/2, 3, 4, 7) for F_n = n-th Fibonacci Number

Then...

((p'_x*p'_2x)*(M_x - (T_x - 1))) / ((T_(M_x) - T_(T_x - 1)) is in N

EXPANSION
((1*1)*(0 + 1))/((0^2 + 0)/2 - (-1^2 -1)/2) = 1
((2*3)*(1 - 0))/((1^2 + 1)/2 - (0^2 + 0)/2) = 6
((3*7)*(3 - 2))/((3^2 + 3)/2 - (2^2 + 2)/2) = 7
((5*13)*(7 - 5)) / ((7^2 + 7)/2 - (5^2 + 5)/2) = 10
((41*101)*(8191 - 90))/((8191^2 + 8191)/2 - (90^2 + 90)/2) = 1

|{-1, 0, 2, 5, 90}| = 1, 0, 2, 5, 90

... where 1, 0, 2, 5, & 90 is the complete set of indices associated with the Ramanujan-Nagell Triangular Numbers...

T_01 = 1
T_00 = 0
T_02 = 3
T_05 = 15
T_90 = 4095

I'm pretty positive these are the only integers for which the above formula holds...

e.g.

((43*135)*(16383 - 104)) / ((16383^2 +16383)/2 - (104^2 + 104)/2) = 645/916 --> .70
((47*176)*(32767 - 119)) / ((32767^2 + 32767)/2 - (119^2 + 119)/2) = 16544/32887 --> .50
((53*231)*(65535 - 135)) / ((65535^2 + 65535)/2 - (135^2 + 135)/2) = 24486/65671 --> .37
((59*297)*(131071 - 152)) / ((131071^2 + 131071)/2 - (152^2 + 152)/2) = 17523/65612 --> 0.26
((61*385)*(262143 - 170)) / ((262143^2 + 262143)/2 - (170^2 + 170)/2) = 23485/131157 --> .18

For the special cases of x = 0, 1, 2 & 13, where par_n denotes "partition #," then the following statement also holds...

((p'_x*par_x)*(M_x - (T_x - 1))) / ((T_(M_x) - T_(T_x - 1)) is in N

Worth noting is the relationship between the following numbers and Multiply Perfect Numbers:

((0^2 + 0)/2 = 0; sigma (0) = 0* (n-fold Perfect)**
((1^2 + 1)/2 = 1; sigma (1) = 1 (1-fold Perfect)
((3^2 + 3)/2 = 6; sigma (6) = 12 (2-fold Perfect)
((7^2 + 7)/2 = 28; sigma (28) = 56 (2-fold Perfect)
((8191^2 + 8191)/2 = 33550336; sigma (33550336) = 67100672 (2-fold Perfect)

* Assumes divisors of an integer must divide that integer and be less than or equal to that integer...
** I don't know if 0 is typically considered "Multiply-Perfect," but I see no reason for it not to be based upon the above definition.

Best,
Raphie

P.S. The above is a "somewhat" accidental observation that followed from exploratory investigations into the following (dimensionless) numerical equivalency for the Josephson Constant Derivation of Planck's Constant where 67092479/8191 = Carol_13/Mersenne_13 = C_13/M_13 for Carol Numbers = M_n^2 - 2, and 9.10938215*10^-31 is the mass of an electron.

sqrt ((4*pi^2*9109.38215)/(67092479/8191) ~ 6.62606776
Related Link: Planck's Constant (Determination) http://en.wikipedia.org/wiki/Planck_...#Determination

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