A245479 Numbers n such that the n-th cyclotomic polynomial has a root mod 7.

1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294, 343, 686, 1029, 2058, 2401, 4802, 7203, 14406, 16807, 33614, 50421, 100842, 117649, 235298, 352947, 705894, 823543, 1647086, 2470629, 4941258, 5764801, 11529602, 17294403, 34588806, 40353607, 80707214, 121060821

Offset: 1

Author

Eric M. Schmidt, Jul 23 2014

Keywords

• nonn
• easy

Comments

Numbers of the form d*7^j for d = 1,2,3,6.

Examples

			The 3rd cyclotomic polynomial x^2 + x + 1 considered modulo 7 has a root x = 2, so 3 is in the sequence.
		

References

• Trygve Nagell, Introduction to Number Theory. New York: Wiley, 1951, pp. 164-168.

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..500
• Benedict W. J. Irwin A sequence a(n+1) is product over binary components of n, plus 1
• Eric Weisstein, Cyclotomic Polynomial.
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,7).

Crossrefs

Cf. A018379, A038754, A245478, A245480, A245481.

Programs

Mathematica

m = 7; Function[d, Table[d[[k]] m^n, {n, 0, 9}, {k, Length@ d}]]@ Divisors[m - 1] // Flatten (* or *)
Rest@ CoefficientList[Series[-x (2 x + 1) (3 x^2 + 1)/(7 x^4 - 1), {x, 0, 40}], x] (* Michael De Vlieger, Jul 25 2016 *)
LinearRecurrence[{0,0,0,7},{1,2,3,6},50] (* Harvey P. Dale, Oct 10 2018 *)

PARI

for(n=1,10^6,if(#polrootsmod(polcyclo(n),7),print1(n,", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */

PARI

Vec(-x*(2*x+1)*(3*x^2+1)/(7*x^4-1) + O(x^100)) \\ Colin Barker, Jul 30 2014

Sage

def A245479(n) : return [6,1,2,3][n%4]*7^((n-1)//4)

Formula

a(n) = 7*a(n-4). G.f.: -x*(2*x+1)*(3*x^2+1) / (7*x^4-1). - Colin Barker, Jul 30 2014

From Benedict W. J. Irwin, Jul 22 2016: (Start)

a(n) appears to satisfy x*Prod_{n>=0} (1 + a(2^n+1)x^(2^n)) = Sum_{n>=1} a(n)x^n.

Then a(n+1)=a(2^x+1)a(2^y+1)a(2^z+1)..., where n=2^x+2^y+2^z+... .

For example,

n=12=2^2+2^3, then a(12+1)=a(2^2+1)*a(2^3+1) i.e. 343=49*7.

n=31=2^0+2^1+2^2+2^3+2^4, then a(31+1)=a(2)*a(3)*a(5)*a(9)*a(17) i.e. 4941258=2*3*7*49*2401.

(End)