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

Author

Christopher J. Smyth, Jan 06 2017

Keywords

• easy
• nonn

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.

Examples

			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).
		

References

• Boyd, David W.(3-BC); Montgomery, Hugh L.(1-MI), Cyclotomic partitions. In Number theory (Banff, AB, 1988), 7-25. Walter de Gruyter & Co., Berlin, 1990 ISBN:3-11-011723-1, MR1106647. [Asymptotics]

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.

Crossrefs

Cf. A014197, 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).

Programs

Mathematica

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

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.

This 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