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.

A138928 Indices n such that A020501(n) = phi(n)(-2) is prime, where Phi is a cyclotomic polynomial.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 24, 26, 30, 31, 32, 34, 38, 39, 40, 43, 45, 49, 54, 56, 61, 62, 63, 66, 75, 79, 80, 85, 87, 98, 101, 117, 120, 122, 127, 130, 138, 154, 161, 167, 170, 178, 183, 184, 186, 187, 191, 192, 199, 205, 207, 208
Offset: 1

Views

Author

M. F. Hasler, Apr 03 2008

Keywords

Comments

While the sequence is not very interesting up to a(n)<300, there are only 4 values in the interval [400,599].

Links

Crossrefs

Cf. A020501, A138920-A138940.

Programs

  • Mathematica
    Select[ Range[3, 1000], PrimeQ[ Cyclotomic[#, -2]] &] (* Robert G. Wilson v, Mar 25 2012 *)
  • PARI
    for( i=1,999, isprime( polcyclo(i,-2)) && print1( i",")) /* for PARI < 2.4.2 use ...subst(polcyclo(i),x,-2)...*/

A019320 Cyclotomic polynomials at x=2.

Original entry on oeis.org

2, 1, 3, 7, 5, 31, 3, 127, 17, 73, 11, 2047, 13, 8191, 43, 151, 257, 131071, 57, 524287, 205, 2359, 683, 8388607, 241, 1082401, 2731, 262657, 3277, 536870911, 331, 2147483647, 65537, 599479, 43691, 8727391, 4033, 137438953471, 174763, 9588151, 61681
Offset: 0

Views

Author

Simon Plouffe

Keywords

Links

Crossrefs

a(n) = A063696(n) - A063698(n) for up to n=104.
Same sequence in binary: A063672.
Cf. A054432, A020501, A034268.

Programs

  • Maple
    with(numtheory,cyclotomic); f := n->subs(x=2,cyclotomic(n,x)); seq(f(i),i=0..64);
  • Mathematica
    Join[{2}, Table[Cyclotomic[n, 2], {n, 1, 40}]] (* Jean-François Alcover, Jun 14 2013 *)
  • PARI
    vector(20,n,polcyclo(n,2)) \\ Charles R Greathouse IV, May 18 2011

Formula

(lcm_{k=1..n} (2^k - 1))/lcm_{k=1..n-1} (2^k - 1), n > 1. - Vladeta Jovovic, Jan 20 2002
Let b(1) = 1 and b(n+1) = lcm(b(n), 2^n-1) then Phi(n,2) = b(n+1)/b(n) = a(n). - Thomas Ordowski, May 08 2013
a(0) = 2; for n > 0, a(n) = (2^n-1)/gcd(a(0)*a(1)*...*a(n-1), 2^n-1). - Thomas Ordowski, May 11 2013

A019325 Cyclotomic polynomials at x=7.

Original entry on oeis.org

7, 6, 8, 57, 50, 2801, 43, 137257, 2402, 117993, 2101, 329554457, 2353, 16148168401, 102943, 4956001, 5764802, 38771752331201, 117307, 1899815864228857, 5649505, 11898664849, 247165843, 4561457890013486057, 5762401, 79797014141614001, 12111126301, 1628413638264057
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Sequence has a(0) = x; see comments in A020501.

Links

Crossrefs

Cf. A019320, A019321, A019322, A019323, A019324.

Programs

  • Maple
    with(numtheory,cyclotomic); f := n->subs(x=7,cyclotomic(n,x)); seq(f(i),i=0..64);
  • Mathematica
    Join[{7}, Cyclotomic[Range[50], 7]] (* Paolo Xausa, Feb 26 2024 *)
  • PARI
    a(n) = if(n==0, 7, polcyclo(n, 7)); \\ Michel Marcus, Dec 16 2017

Extensions

More terms from Michel Marcus, Dec 17 2017

A105603 Sylvester-Jacobsthal cyclotomic numbers.

Original entry on oeis.org

1, 1, 3, 5, 11, 7, 43, 17, 57, 31, 683, 13, 2731, 127, 331, 257, 43691, 73, 174763, 205, 5419, 2047, 2796203, 241, 1016801, 8191, 261633, 3277, 178956971, 151, 715827883, 65537, 1397419, 131071, 24214051, 4033, 45812984491, 524287, 22366891, 61681
Offset: 1

Views

Author

Paul Barry, Apr 15 2005

Keywords

Comments

Primitive parts of Jacobsthal numbers A001045.

Links

Crossrefs

Cf. A020501, A001045, A061446 (with references), A008555.

Formula

a(n)=product{k=1..n-1, if(gcd(n, k)=1, 2+exp(2*pi*I*k/n), 1)}, I=sqrt(-1)
a(n) = (-1)^phi(n)*cyclotomic(n, -2) for n >= 2 (for n = 1 this would be 3), with phi(n) = A000010(n). - Wolfdieter Lang, Jan 19 2015

A020506 Cyclotomic polynomials at x = -7.

Original entry on oeis.org

-7, -8, -6, 43, 50, 2101, 57, 102943, 2402, 117307, 2801, 247165843, 2353, 12111126301, 137257, 6568801, 5764802, 29078814248401, 117993, 1424861898171643, 5649505, 15772610449, 329554457, 3421093417510114543, 5762401, 79787519018560501, 16148168401, 1628413557556843
Offset: 0

Views

Author

Simon Plouffe

Keywords

Comments

Sequence has a(0) = x; see comments in A020501.

Links

Programs

  • Maple
    with(numtheory,cyclotomic); f := n->subs(x=-7,cyclotomic(n,x)); seq(f(i),i=0..64);
  • Mathematica
    Join[{-7}, Cyclotomic[Range[50], -7]] (* Paolo Xausa, Feb 26 2024 *)
  • PARI
    a(n) = if(n==0, -7, polcyclo(n, -7)); \\ Michel Marcus, Dec 17 2017

Extensions

More terms from Michel Marcus, Dec 17 2017
Showing 1-5 of 5 results.

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