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A173263 Successive numbers n such that ChebyshevT[n/2, n] is not an integer. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A296243 A162699 A325129 * 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

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Last modified April 16 12:45 EDT 2021. Contains 343037 sequences. (Running on oeis4.)