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A245479 Numbers n such that the n-th cyclotomic polynomial has a root mod 7. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 27 19:34 EDT 2021. Contains 347694 sequences. (Running on oeis4.)