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A280611 Number of degree n products of distinct cyclotomic polynomials.

%I A280611 #23 May 09 2025 13:58:29
%S A280611 1,2,4,6,10,14,24,34,52,70,102,134,194,254,352,450,610,770,1036,1302,
%T A280611 1716,2130,2770,3410,4406,5402,6892,8382,10600,12818,16120,19422,
%U A280611 24216,29010,35932,42854,52832,62810,76944,91078,111008,130938
%N A280611 Number of degree n products of distinct cyclotomic polynomials.
%C A280611 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.
%D A280611 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]
%H A280611 Vaclav Kotesovec, <a href="/A280611/b280611.txt">Table of n, a(n) for n = 0..10000</a>
%H A280611 L. Kronecker, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002149613">Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten</a>, J. Reine Angew. Math. 53 (1857), 173-175.
%F A280611 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.
%F A280611 This is also the Euler transform of A280712.
%F A280611 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
%e A280611 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).
%t A280611 Table[SeriesCoefficient[Product[(1 + x^EulerPhi@ i), {i, n E^2}], {x, 0, n}], {n, 0, 92}] (* _Michael De Vlieger_, Jan 10 2017 *)
%Y A280611 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).
%K A280611 easy,nonn
%O A280611 0,2
%A A280611 _Christopher J. Smyth_, Jan 06 2017