A280611 Number of degree n products of distinct cyclotomic polynomials.

1, 2, 4, 6, 10, 14, 24, 34, 52, 70, 102, 134, 194, 254, 352, 450, 610, 770, 1036, 1302, 1716, 2130, 2770, 3410, 4406, 5402, 6892, 8382, 10600, 12818, 16120, 19422, 24216, 29010, 35932, 42854, 52832, 62810, 76944, 91078, 111008, 130938

Offset

0,2

Comments

a(n) is also the number monic integer polynomials of degree n all of whose roots are distinct and of modulus 1. This follows from a classical result of Kronecker -- see link.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173-175.

Formula

G.f.: Product_{i>=1} (1 + x^phi(i)) = Product_{j>=1} (1 + x^j)^A014197(j), where phi(i)=A000010(i) is Euler's totient function.

It is also the Euler transform of A280712.

a(n) ~ exp(sqrt(105*zeta(3)*n/2)/Pi) * (105*zeta(3)/2)^(1/4) / (4*Pi*n^(3/4)). - Vaclav Kotesovec, Sep 02 2021

Example

a(3) = 6 because there are six degree-3 products of distinct cyclotomic polynomials, namely (z-1)(z^2+z+1), (z-1)(z^2+1), (z-1)(z^2-z+1), (z+1)(z^2+z+1), (z+1)(z^2+1) and (z+1)(z^2-z+1).

Mathematica

Table[SeriesCoefficient[Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 92}] (* Michael De Vlieger, Jan 10 2017 *)

Crossrefs

Cf. A280709 (variant where z, as well as cyclotomic polynomials, is allowed in the product), A120963 (variant where repeated roots are allowed), A051894 (variant where both z and repeated roots are allowed), A280712 (Inverse Euler transform of sequence).

Sequence in context: A077625 A027383 A364671 * A138016 A239787 A113118

Adjacent sequences: A280608 A280609 A280610 * A280612 A280613 A280614

Keyword

easy,nonn

Author

Christopher J. Smyth, Jan 06 2017

Status

approved