A091813 - OEIS

%I #12 Oct 01 2019 03:57:59

%S 1,1,2,3,3,2,5,6,6,3,7,6,8,5,6,11,11,8,12,10,8,9,15,12,16,10,17,14,17,

%T 8,19,20,13,13,15,23,23,15,17,21,26,11,28,26,24,18,30,23,31,20,21,29,

%U 32,22,25,29,23,23,36,23,37,25,34,39,30,18,41,39,29,22,44,45,45,30,35,44

%N Number of positive squarefree integers k<=n satisfying gcd_*(k,n)=1, where gcd_*(k,n) is the greatest divisor of k that is also a unitary divisor of n.

%D Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7, Unitarism and Infinitarism, pp. 49-56.

%H Amiram Eldar, <a href="/A091813/b091813.txt">Table of n, a(n) for n = 1..10000</a>

%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]

%e a(4)=3 because each of 1, 2, 3 are squarefree and gcd_*(2,4)=1. The latter follows since 2 is not a unitary divisor of 4. a(5)=3 because 4 is not squarefree.

%t udiv[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; uGCD[a_, b_] := Max[Intersection[Divisors[a], udiv[b]]]; a[n_] := Sum[MoebiusMu[k]^2 * Boole[uGCD[k, n] == 1], {k, 1, n}]; Array[a, 76] (* _Amiram Eldar_, Oct 01 2019 *)

%Y Cf. A047994, A073311.

%K nonn

%O 1,3

%A _Steven Finch_, Mar 07 2004