A219277 -id:A219277 - cp's OEIS frontend

A219278 Numbers n such that ChebyshevT[16,n] is prime.

2, 13, 20, 21, 33, 74, 87, 88, 94, 104, 105, 127, 172, 182, 185, 188, 215, 224, 233, 240, 249, 258, 278, 281, 292, 293, 304, 329, 337, 365, 399, 416, 433, 440, 468, 471, 489, 502, 509, 529, 540, 573, 576, 583, 608, 612, 615, 622, 630, 631, 639, 685, 689, 707

Offset: 1

Michel Lagneau, Nov 17 2012

nonn

ChebyshevT[16,x] is the 16th Chebyshev polynomial of the first kind evaluated at x.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A144131, A144132, A219276, A219277.

P:= unapply(orthopoly[T](16,x),x):
select(isprime @ P, [$1..1000]); # Robert Israel, Aug 13 2018

lst={}; Do[If[PrimeQ[ChebyshevT [16, n]], AppendTo[lst, n]], {n, 10^3}]; lst
Select[Range[800],PrimeQ[ChebyshevT[16,#]]&] (* Harvey P. Dale, Jan 23 2016 *)

is(n)=ispseudoprime(polchebyshev(16,1,n)) \\ Charles R Greathouse IV, May 22 2017

A219280 Smallest prime of the form ChebyshevT[2^n, x].

Original entry on oeis.org

2, 7, 97, 665857, 708158977, 150038171394905030432003281854339710977

Offset: 0

Michel Lagneau, Nov 17 2012

nonn

ChebyshevT[2^n, x] is the 2^n th Chebyshev polynomial of the first kind evaluated at x.
The corresponding numbers x are {2, 2, 2, 3, 2, 8, 164, 29, ...}.
a(7) = T(128, 29) = 2518958009…2561281 contains 226 decimal digits.

			T(1, x) = x => a(0) = T(1,2) = 2 ;
T(2, x) = 2x^2 - 1 => a(1) = T(2, 2) = 7 ;
T(4, x) = 8x^4 - 8x^2 + 1 => a(2) = T(4,2) = 97.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 795.
C. W. Jones, J. C. P. Miller, J. F. C. Conn and R. C. Pankhurst, Tables of Chebyshev polynomials, Proc. Roy. Soc. Edinburgh. Sect. A. 62, (1946), 187-203.

Cf. A066436, A144131, A144132, A219276, A219277, A219278, A219279.

Table[k = 0; While[!PrimeQ[ChebyshevT[2^n,k]], k++]; ChebyshevT[2^n,k], {n, 0, 7}]

A219281 Smallest number k such that ChebyshevT[2^n, k] is prime.

Original entry on oeis.org

2, 2, 2, 3, 2, 8, 164, 29, 60, 213, 181, 652

Offset: 0

Michel Lagneau, Nov 17 2012

ChebyshevT[2^n,x] is the 2^n th Chebyshev polynomial of the first kind evaluated at x.

			T(1, x) = x => T(1,2) = 2 is prime => a(0) = 2;
T(2, x) = 2x^2 - 1 => T(2, 2) = 7 is prime => a(1) = 2;
T(4, x) = 8x^4 - 8x^2 + 1 => T(4,2) = 97 is prime => a(2) = 2.

Cf. A066436, A144131, A144132, A219276, A219277, A219278, A219279, A219280.

for n from 0 to 11 do
  P:= unapply(orthopoly[T](2^n,x),x):
  for k from 1 do if isprime(P(k)) then A[n]:= k; break fi od
od:
seq(A[n],n=0..11); # Robert Israel, Aug 13 2018

Table[k = 0; While[!PrimeQ[ChebyshevT[2^n,k]], k++]; k, {n, 0, 7}]

a(10) and a(11) from Robert Israel, Aug 13 2018

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A219278 Numbers n such that ChebyshevT[16,n] is prime.

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A219280 Smallest prime of the form ChebyshevT[2^n, x].

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A219281 Smallest number k such that ChebyshevT[2^n, k] is prime.

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