cp's OEIS Frontend

From Alexander R. Povolotsky, Jun 23 2009, Apr 04 2012: (Start)

One could observe that the last four of Class Number 1 expressions in T. Piezas "Ramanujan Pages" could be expressed as the following approximation:

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

which gives 4 (four) "almost integer" solutions:

1) n = 0, 19+24*0 = 19, k = 4;

2) n = 1, 19+24*1 = 43, k = 40;

3) n = 2, 19+24*2 = 67, k = 220;

4) n = 6, 19+24*6 = 163, k = 26680; this of course is the case for Ramanujan constant vs. its integer counterpart approximation. (End)

From Alexander R. Povolotsky, Oct 16 2010, Apr 04 2012: (Start)

Also if one expands the left part above to exp(Pi*sqrt(b(n))) where b(n) = {19, 25, 43, 58, 67, 163, 232, ...} then the expression (exp(Pi*sqrt(b(n))))/m (where m is either integer 1 or 8) yields values being very close to whole integer value:

Note, that the first differences of b(n) are all divisible by 3, giving after the division: {2, 6, 5, 3, 32, 33, ...}. (End)

From Amiram Eldar, Jun 24 2021: (Start)

This constant was discovered by Hermite (1859).

It is sometimes called "Ramanujan's constant" due to an April Fool's joke by Gardner (1975) in which he claimed that Ramanujan conjectured that this constant is an integer, and that a fictitious "John Brillo" of the University of Arizona proved it on May 1974.

In fact, Ramanujan studied similar near-integers of the form exp(Pi*sqrt(k)) (e.g., A169624), but not this constant.

Gauld (1984) discovered that (Pi*sqrt(163))^e = 22806.9992... is also a near-integer. (End)

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.

Also decimal expansion of log(R_c)/Pi, where R_c is Ramanujan's constant: 262537412640768743.999999999999250... = A060295.

Decimal expansion of k = log(262537412640768744)/Pi.

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

By construction, this constant here is very close to sqrt(163) = A210963.