1,1

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 such that most terms, and in particular record values such as those occuring 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

For links, references and more information see A019296 and other cross-referenced sequences.

a(n)=1/(A056580(n)-e^(sqrt(n)*pi)).

A019296 ={-1, 0} U { n | abs(A056581(n)) >100} U { some n for which abs(A056581(n)) =100 }. - M. F. Hasler, Apr 15 2008

a(6)=110, since e^(pi*sqrt(6))=2197.99087 and 1/(2198-2197.99087)=109.52 which rounds to 110.

e^(pi*sqrt(163))=262537412640768743.99999999999925007259719818 (the Ramanujan number) and so a(163)=1333462407513.

(PARI) default(realprecision, 100); dZ(x)=round(x)-x

A056581(n)=round(1/dZ(exp(sqrt(n)*Pi))

Cf. A003173, A019296-A019297, A035484, A056580, A058292, A060456, A069014, A138851.

Sequence in context: A053155 A041086 A015495 this_sequence A039308 A029525 A037500

Adjacent sequences: A056578 A056579 A056580 this_sequence A056582 A056583 A056584

sign

Henry Bottomley (se16(AT)btinternet.com), Jun 30 2000

Definition, formulae and values corrected and extended by M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Apr 15 2008