It is quite well-known that:

e^(pi*sqrt(19)) ~ 96^3 + 744
e^(pi*sqrt(43)) ~ 960^3 + 744
e^(pi*sqrt(67)) ~ 5280^3 + 744
e^(pi*sqrt(163)) ~ 640320^3 + 744

using the four highest Heegner numbers. But it is not so well-known
that the expression e^(pi*sqrt(d)) can be given *another* internal
structure:

e^(pi*sqrt(19)) ~ 12^3(3^2-1)^3 + 744
e^(pi*sqrt(43)) ~ 12^3(9^2-1)^3 + 744
e^(pi*sqrt(67)) ~ 12^3(21^2-1)^3 + 744
e^(pi*sqrt(163)) ~ 12^3(231^2-1)^3 + 744

The reason for the squares are due to certain Eisenstein series -- but
that's another story.  :-)

Beautifully consistent, aren't they?

I'm working on a new webpage about this and, er, other Ramanujan-
related stuff. But I'm having a devil of a time finishing it due to my
day job. I'll post the link here when it's done.

Yours,

 exp(Pi*sqrt(19+24*n) =~ (24*k)^3 + 31*24

PARI confirmation is below ..

gp > for(n=0,10,print1("n= ",n," k= ",
((exp(Pi*sqrt(19+24*n))/24-31)/24/24)^(1/3),"\n"))

 n= 0 k= 3.999999664954872711861691865  <<========
 n= 1 k= 39.99999999999664632214064072  <<========
n= 2 k= 219.9999999999999993336409313   <<========
n= 3 k= 908.2994607084626509324663895
n= 4 k= 3139.719720204852366879238790
n= 5 k= 9587.574481226312121129336932
n= 6 k= 26680.00000000000000000000000    <<========
n= 7 k= 69020.39408641981880200520050
n= 8 k= 168277.4270764306998213353795
n= 9 k= 390498.9836593266367110562264
n= 10 k= 868910.8509221483459190206684

 Cheers,
Alexander R. Povolotsky

exp(Pi*sqrt(19+24*n) =~ (24*k)^3 + 31*24
gp > for(n=0,10,print1("n= ",n," k= ",
((exp(Pi*sqrt(19+24*n))/24-31)/24/24)^(1/3),"\n"))
n= 0 k=        3.999999664954872711861691865  <<= - 3.9...         0
/ 36   =   0
n= 1 k=      39.99999999999664632214064072 <<=  - 3.9...         36  /
36   =   1
n= 2 k=    219.9999999999999993336409313 <<=  - 3.9...         216  /
36   =    6
n= 6 k= 26680.00000000000000000000000 <<= - 4               26676 /  36   = 741