A173263 Successive numbers n such that ChebyshevT[n/2, n] is not an integer.

3, 5, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 33, 35, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 129, 131, 133, 135, 137

Offset

1,1

Comments

Successive numbers n such that ChebyshevT[x, x/2] is not integer (is integer/2) see A007310

From Robert Israel, Apr 30 2015 (Start)

Numbers 2*k+1 where (1/2)*(2*k+1+2*sqrt(k*(k+1)))^(k+1/2)+(1/2)*(2*k+1+2*sqrt(k*(k+1)))^(-k-1/2) is not an integer.

All odd numbers not in A056220. (End)

From Robert Israel, May 01 2015 (Start)

ChebyshevT(1/2,n) = sqrt((n+1)/2) is an integer iff n is in A056220, otherwise it is irrational.

For odd k, ChebyshevT(k,x) = x*P(x^2) where P is a polynomial with integer coefficients and no roots >= 1.

Therefore if k is odd and n is a positive integer, ChebyshevT(k/2,n) = ChebyshevT(k,ChebyshevT(1/2,n)) = sqrt((n+1)/2)*P((n+1)/2) is an integer iff n is in A056220, otherwise it is irrational. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Maple

remove(t -> issqr((t+1)/2), [seq(2*j+1, j=0..1000)]); # Robert Israel, Apr 30 2015

Mathematica

aa = {}; Do[If[PossibleZeroQ[Round[N[ChebyshevT[x/2, x], 100]] - N[ChebyshevT[x/2, x], 100]], , AppendTo[aa, x]], {x, 0, 1500}]; aa (* Artur Jasinski, Feb 14 2010 *)

Crossrefs

Cf. A007310, A056220.

Sequence in context: A162699 A325129 A368603 * A285141 A329095 A319009

Adjacent sequences: A173260 A173261 A173262 * A173264 A173265 A173266

Keyword

nonn

Author

Artur Jasinski, Feb 14 2010

Extensions

Corrected and edited by Robert Israel, Apr 30 2015

Status

approved