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)

A measure of how close e^(Pi*sqrt(n)) is to an integer (higher absolute value of a(n) means closer, negative value means the closest integer is smaller than it).

The sign convention is chosen so that most terms and in particular record values such as those occurring for the Heegner numbers A003173, are positive, so that A069014 lists record indices of this sequence (except for A069014(2)=2 instead of 3 for signed values). The sequence is not defined for n=0,-1 where e^(sqrt(n)*Pi) is an integer. - M. F. Hasler, Apr 15 2008

Negative resp. positive values of a(n) correspond to 2nd resp. 3rd term of the continued fraction expansion of exp(sqrt(n)*Pi), up to a difference of -1 or -2 depending on the direction of rounding. - M. F. Hasler, Apr 15 2008

Old name of sequence was "Integers that are very close to values of exp(Pi*sqrt(n))", which left "very close" undefined. Robert G. Wilson v resolved this problem on Feb 28 2006 with the comment that "'Very close' means to within 0.01." - Jon E. Schoenfield, Mar 21 2015