This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056581 #16 Jan 06 2025 04:19:00
%S A056581 -7,-51,4,-2,-5,110,15,-3,3,5,-7,-3,19,4,5,-3,430,141,4,4,-2,574,3,7,
%T A056581 1518,-3,62,84,-2,-10,11,-7,-13,-4,4,-3,45551,-5,3,3,2,-33,4494,-8,-5,
%U A056581 -6,3,-2,7,2,9,-3,-4,-4,3,-17,-2,5624716,147,-5,4,3,3,2,6,-2,747638
%N A056581 Nearest integer to 1/(A056580(n) - exp(sqrt(n)*Pi)).
%C A056581 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).
%C A056581 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
%C A056581 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
%D A056581 For links, references and more information see A019296 and other cross-referenced sequences.
%F A056581 a(n) = 1/(A056580(n) - e^(sqrt(n)*Pi)).
%F A056581 A019296 ={-1, 0} U { n | abs(A056581(n)) > 100} U { some n for which abs(A056581(n)) = 100 }. - _M. F. Hasler_, Apr 15 2008
%e A056581 a(6)=110, since e^(Pi*sqrt(6)) = 2197.9908695... and 1/(2198-2197.9908695...) = 109.52... which rounds to 110.
%e A056581 e^(Pi*sqrt(163)) = 262537412640768743.9999999999992500725971981... (the Ramanujan number) and so a(163)=1333462407513.
%o A056581 (PARI) default(realprecision,100); dZ(x)=round(x)-x
%o A056581 A056581(n)=round(1/dZ(exp(sqrt(n)*Pi)))
%Y A056581 Cf. A003173, A019296-A019297, A035484, A056580, A058292, A060456, A069014, A138851.
%K A056581 sign
%O A056581 1,1
%A A056581 _Henry Bottomley_, Jun 30 2000
%E A056581 Definition, formulas and values corrected and extended by _M. F. Hasler_, Apr 15 2008