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A138852 a(n) = round(1/(x-round(x))), where x = (log(744 + (12(n^2-1))^3)/Pi)^2, round(x) = nearest integer to x.

%I A138852 #18 Jan 17 2025 07:23:10
%S A138852 2,-4,1435303,-2,4,4,-6,17,952364958135,-3,4,-2,-5,-8,-7,-4,-2,4,-10,
%T A138852 2,21119108989115042,2,-8,4,-2,-7,10,3,2,-3,-4,-6,-10,-16,-19,-16,-11,
%U A138852 -7,-5,-3,-2,2,3,6,-51,-5,-2,3,7,-10,-3,3,9,-6,-2,5,-9,-2,4,-8,-2,6,-5,2,44,-3,4,-5,3,-35,-2,10,-3,5,-4
%N A138852 a(n) = round(1/(x-round(x))), where x = (log(744 + (12(n^2-1))^3)/Pi)^2, round(x) = nearest integer to x.
%C A138852 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.
%H A138852 Amiram Eldar, <a href="/A138852/b138852.txt">Table of n, a(n) for n = 1..10000</a>
%H A138852 Titus Piezas III, <a href="https://groups.google.com/g/sci.math/c/vYChyV3pA9s/m/Ynv9Xrh5dlwJ">"More on e^(pi*sqrt(163))" on sci.math.research, Apr 13, 2008</a> and his <a href="https://www.oocities.org/titus_piezas/ramanujan.html">Ramanujan Pages</a>.
%e A138852 We have a(3) = 1435303 since (12(3^2-1))^3+744 = e^(pi sqrt(x)) with x = 19.0000006967... = 19 + 1/1435302.833...
%e A138852 In the same way, a(231)=43072298941682041177938098750 since (12(231^2-1))^3+744 = e^(pi sqrt(x)) with x = 163.0000000000000000000000000000232 = 163 + 1/43072298941682041177938098749.8977...
%t A138852 a[n_] := Module[{x = (Log[744 + (12*(n^2-1))^3] / Pi)^2}, Round[1/(x - Round[x])]]; Array[a, 100] (* _Amiram Eldar_, Jan 17 2025 *)
%o A138852 (PARI) default(realprecision,200); A138852(n)={ n=(log(744+(12*(n^2-1))^3)/Pi)^2; round(1/(x-round(x))) }
%Y A138852 Cf. A139388, A138851, A003173, A014708, A056581 and references therein.
%K A138852 sign
%O A138852 1,1
%A A138852 _M. F. Hasler_, Apr 17 2008