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Where exp^(Pi*sqrt163) is the Ramanujan constant and 70^2 is related to the norm vector 0 of the Leech lattice where 1^2 + 2^2 + 3^2 + ... + 22^2 + 23^2 + 24^2 = 70^2. A curiosity is: exp^2(Pi*sqrt163)*70^2 ~ hc/piGm^2 where all physics values are CODATA 2006 and m = neutron mass and exp^2(Pi*sqrt163)*70^2 = 3.377368...x 10^38 and hc/piGm^2 = 3.37700 x 10^38 (+- 0.00050) where 0.00050 = u_c which is the combined standard uncertainty.

This can also be expressed in a symmetric form in terms of the square of the neutron mass in units of Planck mass: where hc/2PiGm^2 = (Mp/m)^2 (Mp = Planck mass and m = neutron mass) and (exp^2(Pi*sqrt163)70^2)/2 ~ (Mp/m)^2. Note the divisor 2 in this case, which yields (exp^2(Pi*sqrt163)*70^2)/2 = 168868437938467735733159816253012231600.00000040115967. - Mark A. Thomas, Jul 02 2009

From Peter Bala, Oct 02 2013: (Start)

A theorem of Kuzmin in the measure theory of continued fractions says that for a random real number alpha, the probability that some given partial quotient of alpha is equal to a positive integer k is given by (1/log(2))*( log(1 + 1/k) - log(1 + 1/(k+1)) ). For example, almost all real numbers have 41.5% of their partial quotients equal to 1, 17% equal to 2, 9.3% equal to 3 and so on. Thus large partial quotients are the exception in continued fraction expansions.

Let now alpha denote the real root of Brillhart's cubic equation x^3 - 8*x - 10 = 0. Then alpha = (5 + (1/9)*sqrt(3*163))^(1/3) + (5 - (1/9)*sqrt(3*163))^(1/3) = 3.31862 82177 50185 65910 .... The continued fraction expansion of alpha is unusual in that there are 8 surprisingly large partial quotients in the first 200 terms, namely [22986, 1501790, 35657, 49405, 53460, 16467250, 48120, 325927]. The explanation for this behavior was given by Stark.

Roberts, p. 227, gives a remarkable product representation for beta := 2 + alpha, namely, beta = q^(1/24) * Product_{n >= 1} (1 + (1/q)^(2*n+1)), where q = exp(Pi*sqrt(163)), and notes that the first factor exp((Pi/24)*sqrt(163)) = 5.31862 82177 50185 63885 ... gives 16 decimal places of beta. Cf. A160514.

Some powers beta^k of beta share with beta the property of having exceptionally large partial quotients early on in their continued fraction expansion. Particularly noteworthy is beta^2, which, like beta, has 8 large partial quotients [126425, 8259853, 1620, 271730, 294038, 90569882, 264667, 1792603] in its first 200 terms. Strangely, apart from the third term 1620, these numbers are almost exactly 5.5 times larger than the corresponding large partial quotients [22986, 1501790, 35657, 49405, 53460, 16467250, 48120, 325927] occurring in the continued fraction expansion of beta.

Also noteworthy is beta^8 = 640320.00000000062437..., very nearly an integer, whose continued fraction expansion begins [640320, 1601600400, 320160, 2135467200, 261949, 10, 1, 20533337, 1, 1, 4, 3, 1, 3, 1369, 2, 2, 14, 3, 9535605, 1, 3, 2, 1, 2, 1, ...].

The cubic irrationals r*beta, where r is rational, also appear to have large partial quotients early on in their continued fraction expansions, which appear to be related to the partial quotients of beta. For example, (2/3)*beta has 8 exceptionally large partial quotients [137919, 9010749, 213951, 296433, 320769, 98803508, 288729, 1955567] in the first 200 terms of its continued fraction expansion and each of these partial quotients is almost exactly 6 times larger than the corresponding large partial quotient in the expansion of beta. (End)

640320^3 + 744 is an integer that is very close to Ramanujan's constant e^(Pi*sqrt(163)) = 640320^3 + 743.99999999999925 and 70^2 is related to the norm vector 0 of the hyperbolic version of the Leech lattice since 1^2 + 2^2 + 3^3 + ... + 22^2 + 23^2 + 24^2 = 70^2.

337736875876935471466319632507953926400 = 2^8*3^2*5^2*7^2*10939058860032031^2 and 10939058860032031 is a prime number that can be decomposed to 2^15*3^2*5^3*23^3*29^3 + 31.

Decimal expansion of log(262537412640768744)/sqrt(163).

First differs from A000796 at a(32).

Note that 262537412640768744 = 24*10939058860032031 = 2^3 * 3 * 10939058860032031, is the nearest integer to the value of Ramanujan's constant e^(Pi*sqrt(163)) = 262537412640768743.999999999999250... = A060295.