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

1,2

Comments

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

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).

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)

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A325094 A191614 A018379 * A294524 A032882 A265394

Adjacent sequences: A245476 A245477 A245478 * A245480 A245481 A245482

Keyword

nonn,easy

Author

Eric M. Schmidt, Jul 23 2014

Status

approved