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 A280611 Number of degree n products of distinct cyclotomic polynomials. 7
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 25 07:41 EDT 2024. Contains 371964 sequences. (Running on oeis4.)