A317272 Numbers k such that Phi(k,x) is a cyclotomic polynomial with middle coefficient different from -1, 0, 1.

385, 770, 1155, 1365, 1540, 1925, 2065, 2145, 2310, 2415, 2465, 2695, 2717, 2730, 2737, 2821, 2849, 3003, 3080, 3255, 3315, 3465, 3619, 3850, 4095, 4130, 4199, 4235, 4277, 4290, 4543, 4620, 4785, 4830, 4845, 4879, 4930, 4991

Offset

1,1

Comments

n is in the sequence if and only if A204455(n) is in the sequence. - Robert Israel, Apr 17 2019

Links

Robert Israel, Table of n, a(n) for n = 1..2000

M. Beiter, The midterm coefficient of the cyclotomic polynomial F_{pq}(x), Amer. Math. Monthly 71 (1964), 769-770.

G. Dresden, On the Middle Coefficient of a Cyclotomic Polynomial, Amer. Math. Monthly 111 (2004), 531-533.

Example

385 is the first item in the list because Phi(385,x) is the first cyclotomic polynomial with middle coefficient different from -1,0,1 (the middle term is -3x^120)

Maple

filter:= proc(n) local p, d;
   p:= numtheory:-cyclotomic(n, x);
   d:= degree(p, x);
   abs(coeff(p, x, d/2))>1
end proc:
select(filter, [$3..5000]); # Robert Israel, Apr 17 2019

Mathematica

Select[Range[3, 4000],
Abs[Coefficient[Cyclotomic[#, x], x, EulerPhi[#]/2]] > 1 &]

Prog

(PARI) isok(n) = (n > 2) && (abs(polcoeff(polcyclo(n), eulerphi(n)/2)) > 1); \\ Michel Marcus, Aug 02 2018

Crossrefs

Cf. A013590, A023022 (phi(n)/2), A204455.

Sequence in context: A013591 A152941 A229853 * A157354 A200525 A225711

Adjacent sequences: A317269 A317270 A317271 * A317273 A317274 A317275

Keyword

nonn

Author

Greg Dresden, Jul 25 2018

Status

approved