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

5,1

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.

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.

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A117462 A155462 A109496 * A181061 A329934 A269611

Adjacent sequences: A138848 A138849 A138850 * A138852 A138853 A138854

Keyword

sign,changed

Author

M. F. Hasler, Apr 16 2008

Status

approved