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

-4, -3, -2, 2, 3, 5, 12, -33, -7, -4, -2, 2, 3, 6, 8954018, -6, -3, 2, 3, 9, -12, -3, -2, 4, 18, -6, -2, 3, 14, -5, -2, 4, -21, -3, 3, 51, -3, 3, 2683620901418, -3, 4, -9, 2, 11, -3, 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

Offset: 5

Author

M. F. Hasler, Apr 16 2008

Keywords

• sign

Comments

Records are attained at the larger Heegener numbers (A003173).

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.

Examples

			We have e^(Pi sqrt(19))-744 = (12(x^2-1))^3 with x = 2.9999998883... = 3 - 1/8954017.533..., therefore a(19) = 8954018.
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 5..10000
• Titus Piezas III, "More on e^(pi*sqrt(163))" on sci.math.research, April 13, 2008 and The Ramanujan Pages.

Crossrefs

Cf. A003173, A014708, A056581 and references therein, A138852.

Programs

Mathematica

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 *)

PARI

default(realprecision,200); A138851(n)={ n=frac( sqrt( sqrtn( exp( sqrt(n)*Pi )-744, 3)/12 + 1 )); round( 1/(round(n)-n)) }