A105603 -id:A105603 - cp's OEIS frontend

A020501 Cyclotomic polynomials at x=-2.

-2, -3, -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

Offset: 0

Simon Plouffe

sign

a(0) depends on the definition of the 0th cyclotomic polynomial; Maple defines it as x, but Mathematica defines it as 1. - T. D. Noe, Jul 23 2008 [a(0) = x is correct. - N. J. A. Sloane, Aug 01 2008]

A020501[2n] = A019320[n] for all odd n > 1. (Because if m > 1 is odd, then Phi_2m(x) = Phi_m(-x) as demonstrated by Bloom). - Antti Karttunen, Aug 02 2001

T. D. Noe, Table of n, a(n) for n = 0..1000
D. M. Bloom, On the Coefficients of the Cyclotomic Polynomials, Amer.Math.Monthly 75, 372-377, 1968.
Index entries for cyclotomic polynomials, values at X

Cf. A020500, A020513.

Cf. A105603

with(numtheory,cyclotomic); f := n->subs(x=-2,cyclotomic(n,x)); seq(f(i),i=0..64);

Join[{-2}, Cyclotomic[Range[50], -2]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = if (n, polcyclo(n, -2), -2); \\ Michel Marcus, Mar 05 2016

A253807 Primitive part of A006190(n), n >= 1.

Original entry on oeis.org

1, 3, 10, 11, 109, 12, 1189, 119, 1297, 131, 141481, 118, 1543321, 1429, 15445, 14159, 183642229, 1299, 2003229469, 14041, 1837837, 170039, 238367471761, 14158, 23854956949, 1854841, 2186871697, 1670761, 309400794703549

Offset: 1

Wolfdieter Lang, Jan 19 2015

A006190(n) = Product_{k divides n} a(k), n >= 1.

Cf. A000010, A006190, A008683, A013595.

Cf. A061446, A008555, A105603.

(* b = A006190 *) b[0] = 0; b[1] = 1; b[n_] := b[n] = 3*b[n-1] + b[n-2]; a[n_] := Product[b[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Jan 20 2015 *)

a(n) = ((3-sqrt(13))/2)^phi(n)*cyclotomic(n, -(11 - 3*sqrt13)/2) for n >= 1 and a(1) = 1, where phi is Euler's totient A000010 and the coefficient table for the cyclotomic polynomials is given in A013595.

a(n) = Product_{d|n} A006190(d)^mu(n/d), where mu = A008683, n >= 1.

A105604 Sylvester dividends for Jacobsthal numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 5, 3, 11, 1, 105, 1, 43, 33, 85, 1, 1197, 1, 1705, 129, 683, 1, 23205, 11, 2731, 171, 27305, 1, 2370291, 1, 21845, 2049, 43691, 473, 5679765, 1, 174763, 8193, 5941925, 1, 621456339, 1, 6990505, 622611, 2796203, 1, 1437248085, 43, 346729141, 131073, 111848105, 1, 22861753173, 7513, 1521134245

Offset: 1

Paul Barry, Apr 15 2005

Divide each Jacobsthal number by its primitive part.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A001045, A105603.

A001045(n) = ((2^n - ((-1)^n)) / 3);
A105603(n) = if(1==n,n,(-1)^eulerphi(n)*polcyclo(n, -2)); \\ After Wolfdieter Lang's formula in that entry.
A105604(n) = (A001045(n)/A105603(n)); \\ Antti Karttunen, Nov 01 2017

a(n) = A001045(n)/A105603(n).

More terms from Antti Karttunen, Nov 01 2017

cp's OEIS Frontend

A020501 Cyclotomic polynomials at x=-2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A253807 Primitive part of A006190(n), n >= 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A105604 Sylvester dividends for Jacobsthal numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions