cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A219276 Numbers n such that T_4(n) is prime, where T_4(x) = 8x^4 - 8x^2 + 1 is the fourth Chebyshev polynomial (of the first kind).

Original entry on oeis.org

2, 3, 5, 8, 10, 14, 17, 19, 31, 32, 34, 35, 39, 48, 50, 51, 53, 54, 59, 61, 73, 76, 78, 84, 88, 90, 97, 101, 102, 105, 107, 110, 121, 126, 128, 134, 135, 139, 143, 144, 146, 152, 153, 158, 167, 171, 172, 178, 180, 184, 186, 187, 189, 201, 202, 203, 205, 207
Offset: 1

Views

Author

Michel Lagneau, Nov 17 2012

Keywords

Comments

The corresponding primes are in A144131.
Sequence is infinite under Bunyakovsky's conjecture. - Charles R Greathouse IV, May 29 2013

Links

Crossrefs

Cf. A144131.

Programs

  • Maple
    filter:= n -> isprime(8*n^4 - 8*n^2+1):
select(filter, [$1..300]); # Robert Israel, Jan 22 2020
  • Mathematica
    lst={}; Do[If[PrimeQ[ChebyshevT [4, n]], AppendTo[lst, n]], {n, 10^3}]; lst
  • PARI
    is(n)=isprime(polchebyshev(4,1,n)) \\ Charles R Greathouse IV, May 29 2013

A219277 Numbers n such that ChebyshevT[8,n] is prime.

Original entry on oeis.org

3, 4, 7, 15, 18, 19, 37, 43, 46, 47, 62, 74, 75, 84, 89, 90, 92, 96, 105, 112, 130, 139, 158, 163, 182, 189, 190, 202, 213, 217, 218, 225, 233, 255, 256, 271, 280, 288, 293, 301, 314, 317, 329, 335, 337, 349, 350, 354, 360, 364, 365, 368, 376, 396, 416, 422
Offset: 1

Views

Author

Michel Lagneau, Nov 17 2012

Keywords

Comments

ChebyshevT[8,x] is the 8th Chebyshev polynomial of the first kind evaluated at x.
The corresponding primes are in A144132.

Links

Crossrefs

Cf. A144131, A144132.

Programs

  • Mathematica
    lst={}; Do[If[PrimeQ[ChebyshevT [8, n]], AppendTo[lst, n]], {n, 10^3}]; lst
Select[Range[500],PrimeQ[ChebyshevT[8,#]]&] (* Harvey P. Dale, Jul 23 2025 *)
  • PARI
    is(n)=ispseudoprime(polchebyshev(8,1,n)) \\ Charles R Greathouse IV, May 22 2017

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

Original entry on oeis.org

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

Views

Author

Michel Lagneau, Nov 17 2012

Keywords

Comments

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

Links

Crossrefs

Cf. A144131, A144132, A219276, A219277.

Programs

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

Views

Author

Michel Lagneau, Nov 17 2012

Keywords

Comments

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.

Examples

			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.

References

  • 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.

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Michel Lagneau, Nov 17 2012

Keywords

Comments

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

Examples

			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.

Crossrefs

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

Programs

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

Extensions

a(10) and a(11) from Robert Israel, Aug 13 2018
Showing 1-5 of 5 results.

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