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Ramanujan's constant

Ramanujan's constant is[1]

${\displaystyle e^{-i\pi {\sqrt {-163}}}=e^{\pi {\sqrt {163}}}=({\rm {Gelfond's~constant}})^{\sqrt {163}},\,}$
where Gelfond's constant is
 e π
.
Ramanujan's constant is amazingly close to an integer, the first
 12
digits after the decimal point being
 9
.

Decimal expansion of Ramanujan's constant

The decimal expansion of Ramanujan's constant is (this is almost an integer!)

${\displaystyle e^{\pi {\sqrt {163}}}=262537412640768743.9999999999992500725971981856888793538563373369908627075374103782\ldots \,}$
A060295 Decimal expansion of
 e π 2√  163
.
 {2, 6, 2, 5, 3, 7, 4, 1, 2, 6, 4, 0, 7, 6, 8, 7, 4, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 5, 0, 0, 7, 2, 5, 9, 7, 1, 9, 8, 1, 8, 5, 6, 8, 8, 8, 7, 9, 3, 5, 3, 8, 5, 6, 3, 3, 7, 3, 3, 6, 9, 9, 0, 8, 6, 2, 7, 0, 7, 5, 3, 7, 4, 1, 0, ...}

Decimal expansion of a close approximation to Ramanujan's constant

Ramanujan's constant can be approximated to
 14
digits after the decimal point by the first root[2] of the
 24
th degree polynomial
${\displaystyle (x^{3}-6x^{2}+4x-2)_{1}^{8}\,-\,24\approx 262537412640768743.999999999999251123875936799800954417367910227716\ldots \,}$

A102912 Decimal expansion of a close approximation to the Ramanujan constant.

 {2, 6, 2, 5, 3, 7, 4, 1, 2, 6, 4, 0, 7, 6, 8, 7, 4, 3, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 2, 5, 1, 1, 2, 3, 8, 7, 5, 9, 3, 6, 7, 9, 9, 8, 0, 0, 9, 5, 4, 4, 1, 7, 3, 6, 7, 9, 1, 0, 2, 2, 7, 7, 1, 6, 6, 3, 5, 3, 5, 7, 0, 9, 1, 7, 6, 1, ...}

Continued fraction for Ramanujan's constant

The simple continued fraction for Ramanujan's constant is

${\displaystyle e^{\pi {\sqrt {163}}}=262537412640768743\,+\,{\cfrac {1}{1+{\cfrac {1}{1333462407511+{\cfrac {1}{1+{\cfrac {1}{8+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{5+{\cfrac {1}{1+{\cfrac {1}{4+{\cfrac {1}{1+{\cfrac {1}{\ddots }}}}}}}}}}}}}}}}}}}}}}\,}$
A058292 Continued fraction for
 e π 2√  163
.
 {262537412640768743, 1, 1333462407511, 1, 8, 1, 1, 5, 1, 4, 1, 7, 1, 1, 1, 9, 1, 1, 2, 12, 4, 1, 15, 4, 299, 3, 5, 1, 4, 5, 5, 1, 28, 3, 1, 9, 4, 1, 6, 1, 1, 1, 1, 1, 1, 51, 11, 5, 3, 2, 1, 1, 1, 1, 2, 1, 5, 1, 9, 1, ...}

Sequences

A019297 Integers that are very close to values of
 e π 2√  n
.
 {–1, 1, 2198, 422151, 614552, 2508952, 6635624, 199148648, 884736744, 24591257752, 30197683487, 147197952744, 545518122090, 70292286279654, 39660184000219160, 45116546012289600, ...}
A019296 Values of
 n
for which
 e π 2√  n
is very close to an integer.
 {–1, 0, 6, 17, 18, 22, 25, 37, 43, 58, 59, 67, 74, 103, 148, 149, 163, 164, 177, 205, 223, 226, 232, 267, 268, 326, 359, 386, 522, 566, 630, 638, 652, 719, 790, 792, 928, 940, 986, 1005, 1014, 1169, 1194, ...}
For
 −1
, corresponding to
 n = −1
, it is not only very close, it is exact
${\displaystyle -1=e^{\pi {\sqrt {-1}}}\,}$
which gives the special case of Euler's formula that connects the five most common constants of mathematics:
 e, π, i, 0,
and
 1
${\displaystyle e^{\pi i}+1=0.\,}$
For
 1
, corresponding to
 n = 0
, it is not only very close, it is exact, but in a trivial way!
A178449 Conjectured expansion of
 e π 2√  163
in powers of
 t
, where
 t = 640320  − 3
.
 {1, 744, –196884, 167975456, –180592706130, 217940004309743, –19517553165954887, 74085136650518742, –131326907606533204, ...}