A091813 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.

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, 8, 19, 20, 13, 13, 15, 23, 23, 15, 17, 21, 26, 11, 28, 26, 24, 18, 30, 23, 31, 20, 21, 29, 32, 22, 25, 29, 23, 23, 36, 23, 37, 25, 34, 39, 30, 18, 41, 39, 29, 22, 44, 45, 45, 30, 35, 44

Offset: 1

Author

Steven Finch, Mar 07 2004

Keywords

• nonn

Examples

			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.
		

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Crossrefs

Cf. A047994, A073311.

Programs

Mathematica

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 *)