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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.
2

%I #36 Mar 01 2023 04:42:28

%S 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,

%T 24,27,26,29,26,30,31,32,31,35,35,36,35,39,38,41,40,41,43,45,42,46,46,

%U 48,48,51,49,52,51,54,55,57,55,59,59,59,57,62,62,65,64,66,66,69,66,71

%N 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.

%C 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

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

%H Robert Israel, <a href="/A075795/b075795.txt">Table of n, a(n) for n = 1..10000</a>

%H T. M. Apostol, <a href="http://dx.doi.org/10.1090/S0002-9939-1970-0251010-X">Resultants of Cyclotomic Polynomials</a>, Proc. Amer. Math. Soc. 24, 457-462, 1970.

%H T. M. Apostol, <a href="http://www.jstor.org/stable/2005456">The Resultant of the Cyclotomic Polynomials Fm(ax) and Fn(bx)</a>, Math. Comput. 29, 1-6, 1975.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic polynomials</a>.

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

%F a(n) = n - A001222(n) - 1. - _Michel Marcus_, Jul 24 2016

%p seq(n -numtheory:-bigomega(n)-1, n=1..1000); # _Robert Israel_, Jul 25 2016

%t Table[n - PrimeOmega@ n - 1, {n, 73}] (* _Michael De Vlieger_, Jul 26 2016 *)

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

%Y Cf. A001222, A054372, A073093.

%K nonn

%O 1,5

%A _Benoit Cloitre_, Oct 13 2002

%E a(30)=2 and a(31)=6 merged into a(30)=26 by _Mats Granvik_, Jul 24 2016