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A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.

%I A373318 #9 Jun 01 2024 04:03:07
%S A373318 1,1,2,1,4,1,6,1,2,1,10,1,12,3,8,1,16,1,18,1,4,5,22,1,4,3,2,3,28,2,30,
%T A373318 1,20,4,24,1,36,9,8,1,40,1,42,5,8,11,46,1,6,1,32,3,52,1,8,3,4,7,58,1,
%U A373318 60,15,4,1,48,5,66,2,44,6,70,1,72,9,8,9,60,2,78
%N A373318 Numerator of the asymptotic density of numbers that are unitarily divided by n.
%C A373318 Numbers that are unitarily divided by n are numbers k such that n is a unitary divisor of k, or equivalently, numbers of the form m*n, with gcd(m, n) = 1.
%H A373318 Amiram Eldar, <a href="/A373318/b373318.txt">Table of n, a(n) for n = 1..10000</a>
%H A373318 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>.
%H A373318 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unitary_divisor">Unitary divisor</a>.
%F A373318 a(n) = 1 if and only if n is in A090778.
%F A373318 a(n) = A000010(n) if and only if n is a cyclic number (A003277).
%F A373318 Let f(n) = a(n)/A373319(n). Then:
%F A373318 f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).
%F A373318 f(n) = A173557(n)/A064549(n).
%F A373318 f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.
%F A373318 Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).
%e A373318 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 A373318 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) = numerator(1/4) = 1.
%t A373318 a[n_] := Numerator[EulerPhi[n]/n^2]; Array[a, 100]
%o A373318 (PARI) a(n) = numerator(eulerphi(n)/n^2);
%Y A373318 Cf. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.
%Y A373318 Cf. A001620, A013661, A306016.
%Y A373318 Numbers that are unitarily divided by k: A000027 (k=1), A016825 (k=2), A016051 (k=3), A017113 (k=4), A051062 (k=8), A051063 (k=9).
%K A373318 nonn,easy,frac
%O A373318 1,3
%A A373318 _Amiram Eldar_, Jun 01 2024