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A280709 The number of monic integer polynomials of degree n all of whose roots are distinct and of modulus at most 1. 3
1, 3, 6, 10, 16, 24, 38, 58, 86, 122, 172, 236, 328, 448, 606, 802, 1060, 1380, 1806, 2338, 3018, 3846, 4900, 6180, 7816, 9808, 12294, 15274, 18982, 23418, 28938, 35542, 43638, 53226, 64942, 78786, 95686, 115642, 139754, 168022, 202086, 241946 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Such polynomials are a product of distinct cyclotomic polynomials, possibly multiplied by z. This follows from a classical result of Kronecker -- see Links.

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

a(0) = 1 and a(n) = b(n)+b(n-1) for n >= 1, where b(n) = A280611(n).

G.f.: (1+x)*Product_{i>=1} (1+x^phi(i)) = (1+x)*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 except with its first two terms (2,1) replaced by (3,0).

EXAMPLE

a(2)=6 because the six polynomials z^2+z+1, z^2+1, z^2-z+1, z^2-z, z^2+z and z^2-1 are the only ones of the required type.

MATHEMATICA

Table[SeriesCoefficient[(1 + x) Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 120}] (* Michael De Vlieger, Jan 10 2017 *)

CROSSREFS

Cf. A280611 (variant where all roots must have modulus exactly 1);

Cf. A120963 (variant where multiple roots are allowed).

Sequence in context: A122046 A078663 A173691 * A025222 A011902 A025004

Adjacent sequences:  A280706 A280707 A280708 * A280710 A280711 A280712

KEYWORD

easy,nonn

AUTHOR

Christopher J. Smyth, Jan 07 2017

STATUS

approved

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Last modified August 14 13:55 EDT 2020. Contains 336481 sequences. (Running on oeis4.)