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

1, 4, 9, 8, 25, 18, 49, 16, 27, 25, 121, 36, 169, 98, 225, 32, 289, 54, 361, 50, 147, 242, 529, 72, 125, 169, 81, 196, 841, 225, 961, 64, 1089, 289, 1225, 108, 1369, 722, 507, 100, 1681, 147, 1849, 484, 675, 1058, 2209, 144, 343, 125, 2601, 338, 2809, 162, 605

Offset: 1

Author

Amiram Eldar, Jun 01 2024

Keywords

• nonn
• easy
• frac

Examples

			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, ...
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Unitary Divisor.
• Wikipedia, Unitary divisor.

Crossrefs

Cf. A003277, A373318 (numerators), A373320.

Programs

Mathematica

a[n_] := Denominator[EulerPhi[n]/n^2]; Array[a, 100]

PARI

a(n) = denominator(eulerphi(n)/n^2);

PARI

for(n=1, 100, print1(denominator(direuler(p=2, n, (1-X/p^2)/(1-X/p))[n]), ", ")) \\ Vaclav Kotesovec, Jun 01 2024

Formula

a(n) = n^2 if and only if n is a cyclic number (A003277).