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A373319 Denominator of the asymptotic density of numbers that are unitarily divided by n.

%I A373319 #10 Jun 01 2024 04:03:11
%S A373319 1,4,9,8,25,18,49,16,27,25,121,36,169,98,225,32,289,54,361,50,147,242,
%T A373319 529,72,125,169,81,196,841,225,961,64,1089,289,1225,108,1369,722,507,
%U A373319 100,1681,147,1849,484,675,1058,2209,144,343,125,2601,338,2809,162,605
%N A373319 Denominator of the asymptotic density of numbers that are unitarily divided by n.
%H A373319 Amiram Eldar, <a href="/A373319/b373319.txt">Table of n, a(n) for n = 1..10000</a>
%H A373319 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.
%H A373319 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%F A373319 a(n) = n^2 if and only if n is a cyclic number (A003277).
%e A373319 Fractions begin with: 1, 1/4, 2/9, 1/8, 4/25, 1/18, 6/49, 1/16, 2/27, 1/25, 10/121, 1/36, ...
%e A373319 For n = 2, the numbers that are unitarily divided by 2 are the numbers of the form 4*k+2 whose asymptotic density is 1/4. Therefore a(2) = denominator(1/4) = 4.
%t A373319 a[n_] := Denominator[EulerPhi[n]/n^2]; Array[a, 100]
%o A373319 (PARI) a(n) = denominator(eulerphi(n)/n^2);
%o A373319 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, (1-X/p^2)/(1-X/p))[n]), ", ")) \\ _Vaclav Kotesovec_, Jun 01 2024
%Y A373319 Cf. A003277, A373318 (numerators), A373320.
%K A373319 nonn,easy,frac
%O A373319 1,2
%A A373319 _Amiram Eldar_, Jun 01 2024