A088306 Integers n with tan n > |n|, ordered by |n|.

1, -2, -11, -33, -52174, 260515, -573204, 37362253, -42781604, 122925461, 534483448, 3083975227, 902209779836, -2685575996367, -65398140378926, 74357078147863, 214112296674652, 642336890023956, -5920787228742393, -12055686754159438, 18190586279576483, -48436859313312404

Offset: 1

Author

Paul Boddington, Nov 05 2003

Keywords

• sign

Comments

Name was "Positive integers n with |tan n| > n." before signs were added. The sign here shows whether tan(|n|) is positive or negative.

That this sequence is infinite was proved by Bellamy, Lagarias and Lazebnik. It seems not to be known whether there are infinitely many n with tan n > n.

At approximately 2.37e154, there is a value of n which has tan(n)/n > 556. - Phil Carmody, Mar 04 2007 [This is index 214 in the b-file.]

As n increases, log(|a(n)|)/n seems to approach Pi/2; this is similar to what would be expected if an integer sequence were created by drawing many random numbers independently from a uniform distribution on the interval [-Pi/2,+Pi/2] and including in the sequence only those integers j for which the j-th random number x_j happened to satisfy |x_j| < 1/j (and applying to j the sign of x_j). - Jon E. Schoenfield, Aug 19 2014; updated Nov 07 2014 to reflect the change in the sequence's Name)

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..1000
• D. Bellamy, J. C. Lagarias and F. Lazebnik, Proposed problem: large values of Tan n
• Jon E. Schoenfield, Magma program

Crossrefs

Cf. A000503, A224269, A079330, A088989.

Cf. A249836 (subsequence of positive terms).

Programs

Maple

a:=proc(n) if abs(evalf(tan(n)))>n then n else fi end: seq(a(n),n=1..100000); # Emeric Deutsch, Dec 18 2004

Mathematica

Select[Range[600000],Abs[Tan[#]]>#&] (* Harvey P. Dale, Nov 30 2012 *)

PARI

is(n)=tan(n)>abs(n) \\ Charles R Greathouse IV, Nov 07 2014

Extensions

More terms from Jon E. Schoenfield, Aug 17 2014

Signs added and other edits by Franklin T. Adams-Watters, Sep 09 2014