A245478 Numbers k such that the k-th cyclotomic polynomial has a root mod 5.

1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500, 625, 1250, 2500, 3125, 6250, 12500, 15625, 31250, 62500, 78125, 156250, 312500, 390625, 781250, 1562500, 1953125, 3906250, 7812500, 9765625, 19531250, 39062500, 48828125, 97656250, 195312500, 244140625, 488281250

Offset

1,2

Comments

Numbers of the form 2^i * 5^j for 0 <= i <= 2 and j >= 0.

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

Eric Weisstein, Cyclotomic Polynomial.

Index entries for linear recurrences with constant coefficients, signature (0,0,5).

Formula

a(3j + i) = 2^(i-1)*5^j for i = 1,2,3 and j >= 0.

a(n) = 5*a(n-3). G.f.: -x*(4*x^2+2*x+1) / (5*x^3-1). - Colin Barker, Aug 01 2014

Example

The 4th cyclotomic polynomial x^2 + 1 considered modulo 5 has a root x = 2, so 4 is in the sequence.

Prog

(Sage) def A245478(n) : return 2^((n-1)%3)*5^((n-1)//3)
(PARI) for(n=1, 10^6, if(#polrootsmod(polcyclo(n), 5), print1(n, ", "))) /* by definition; rather inefficient. - Joerg Arndt, Jul 28 2014 */
(PARI) is(n)=n%8 && 2^valuation(n, 2)*5^valuation(n, 5)==n \\ Charles R Greathouse IV, Jul 29 2014
(PARI) Vec(-x*(4*x^2+2*x+1)/(5*x^3-1) + O(x^100)) \\ Colin Barker, Aug 01 2014

Crossrefs

Cf. A000079, A038754, A245479, A245480, A245481.

Sequence in context: A018283 A249398 A018489 * A018530 A018554 A293538

Adjacent sequences: A245475 A245476 A245477 * A245479 A245480 A245481

Keyword

nonn,easy

Author

Eric M. Schmidt, Jul 23 2014

Status

approved