A075795 Number of k, 0<k<=n, such that the resultant of the k-th cyclotomic polynomial and the n-th cyclotomic polynomial is equal to 1.

0, 0, 1, 1, 3, 3, 5, 4, 6, 7, 9, 8, 11, 11, 12, 11, 15, 14, 17, 16, 18, 19, 21, 19, 22, 23, 23, 24, 27, 26, 29, 26, 30, 31, 32, 31, 35, 35, 36, 35, 39, 38, 41, 40, 41, 43, 45, 42, 46, 46, 48, 48, 51, 49, 52, 51, 54, 55, 57, 55, 59, 59, 59, 57, 62, 62, 65, 64, 66, 66, 69, 66, 71

Offset

1,5

Comments

a(n) >= A000010(n)-1 since if 2<=k<n and (k,n)=1, the resultant is 1. - corrected by Robert Israel, Jul 24 2016

For n>1 a(n) = number of roots of the n-th polynomial in A275345, equal to 1. - Mats Granvik, Jul 24 2016

Links

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

T. M. Apostol, Resultants of Cyclotomic Polynomials, Proc. Amer. Math. Soc. 24, 457-462, 1970.

T. M. Apostol, The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx), Math. Comput. 29, 1-6, 1975.

Eric Weisstein's World of Mathematics, Cyclotomic polynomials.

Formula

a(n) = n - A073093(n).

a(n) = n - A001222(n) - 1. - Michel Marcus, Jul 24 2016

Maple

seq(n -numtheory:-bigomega(n)-1, n=1..1000); # Robert Israel, Jul 25 2016

Mathematica

Table[n - PrimeOmega@ n - 1, {n, 73}] (* Michael De Vlieger, Jul 26 2016 *)

Prog

(PARI) a(n)=sum(k=1, n, if(1-polresultant(polcyclo(n), polcyclo(k)), 0, 1))

Crossrefs

Cf. A001222, A054372, A073093.

Sequence in context: A116192 A090104 A333822 * A058268 A341729 A087851

Adjacent sequences: A075792 A075793 A075794 * A075796 A075797 A075798

Keyword

nonn

Author

Benoit Cloitre, Oct 13 2002

Extensions

a(30)=2 and a(31)=6 merged into a(30)=26 by Mats Granvik, Jul 24 2016

Status

approved