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%I #12 Apr 11 2021 16:11:44
%S -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 1518,-3,62,84,-2,-10,11,-7,-13,-4,4,-3,45551,-5,3,3,2,-33,4494,-8,-5,
%U -6,3,-2,7,2,9,-3,-4,-4,3,-17,-2,5624716,147,-5,4,3,3,2,6,-2,747638
%N Nearest integer to 1/(A056580(n) - exp(sqrt(n)*Pi)).
%C 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 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 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 For links, references and more information see A019296 and other cross-referenced sequences.
%F a(n) = 1/(A056580(n) - e^(sqrt(n)*Pi)).
%F 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 a(6)=110, since e^(Pi*sqrt(6)) = 2197.9908695... and 1/(2198-2197.9908695...) = 109.52... which rounds to 110.
%e e^(Pi*sqrt(163)) = 262537412640768743.9999999999992500725971981... (the Ramanujan number) and so a(163)=1333462407513.
%o (PARI) default(realprecision,100); dZ(x)=round(x)-x
%o A056581(n)=round(1/dZ(exp(sqrt(n)*Pi))
%Y Cf. A003173, A019296-A019297, A035484, A056580, A058292, A060456, A069014, A138851.
%K sign
%O 1,1
%A _Henry Bottomley_, Jun 30 2000
%E Definition, formulas and values corrected and extended by _M. F. Hasler_, Apr 15 2008
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