cp's OEIS Frontend

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

Related to almost-integer values of e^(pi sqrt n), obtained for larger Heegener numbers (A003173): T. Piezas draws attention on the fact that the well-known integers very close to exp(pi sqrt(n)) are of the form (12(k^2-1))^3+744. Here this is expressed as the (rounded value) of the reciprocal of the (signed) distance from the integers of the n-value corresponding to a given integer k-value. As expected, records are obtained for k = 3, 9, 21, 231.

The well-known integers very close to e^(pi sqrt n) for n=A003173(m) are of the form (12(k^2-1))^3+744. This sequence gives the rounded n-value corresponding to a given k.