cp's OEIS Frontend

A138851 Nearest integer to 1/(round(x)-x), where exp(Pi sqrt(n))-744 = (12(x^2-1))^3.

%I A138851 #15 Jan 17 2025 03:36:41
%S A138851 -4,-3,-2,2,3,5,12,-33,-7,-4,-2,2,3,6,8954018,-6,-3,2,3,9,-12,-3,-2,4,
%T A138851 18,-6,-2,3,14,-5,-2,4,-21,-3,3,51,-3,3,2683620901418,-3,4,-9,2,11,-3,
%U A138851 4,-5,3,-10,2,-17,2,-14,2,-7,3,-4,7,-2,-16,3,-3,31514540715033062,3,-3,-12,5,2,-3,-9,12,4,2,-2,-3,-4,-7,-10,-16,-19,-16
%N A138851 Nearest integer to 1/(round(x)-x), where exp(Pi sqrt(n))-744 = (12(x^2-1))^3.
%C A138851 Records are attained at the larger Heegener numbers (A003173).
%C A138851 T. Piezas draws attention on the fact that the integers very close to exp(pi sqrt(n)) are of the form (12(k^2-1))^3+744. Here this closeness is expressed as the (rounded value) of the reciprocal of the (signed) distance of these k-values from the integers.
%H A138851 Amiram Eldar, <a href="/A138851/b138851.txt">Table of n, a(n) for n = 5..10000</a>
%H A138851 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, April 13, 2008</a> and <a href="https://www.oocities.org/titus_piezas/ramanujan.html">The Ramanujan Pages</a>.
%e A138851 We have e^(Pi sqrt(19))-744 = (12(x^2-1))^3 with x = 2.9999998883... = 3 - 1/8954017.533..., therefore a(19) = 8954018.
%e A138851 In the same way, e^(Pi sqrt(163))-744 = (12(x^2-1))^3 with x = 230.999999999999999999999999999890... = 231 - 1/9093255353570474976233448828.20..., thus a(163) = 9093255353570474976233448828.
%t A138851 a[n_] := Module[{x = Sqrt[Surd[Exp[Pi * Sqrt[n]] - 744, 3] / 12 + 1]}, Round[1/(Round[x] - x)]]; Array[a, 100, 5] (* _Amiram Eldar_, Jan 17 2025 *)
%o A138851 (PARI) default(realprecision,200); A138851(n)={ n=frac( sqrt( sqrtn( exp( sqrt(n)*Pi )-744, 3)/12 + 1 )); round( 1/(round(n)-n)) }
%Y A138851 Cf. A003173, A014708, A056581 and references therein, A138852.
%K A138851 sign
%O A138851 5,1
%A A138851 _M. F. Hasler_, Apr 16 2008