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

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, 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, 60, 15, 4, 1, 48, 5, 66, 2, 44, 6, 70, 1, 72, 9, 8, 9, 60, 2, 78

Offset: 1

Author

Amiram Eldar, Jun 01 2024

Keywords

• nonn
• easy
• frac

Comments

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.

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) = numerator(1/4) = 1.
		

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. A000010, A003277, A034444, A064549, A076512, A077610, A109395, A173557, A373319(denominators), A373320.

Cf. A001620, A013661, A306016.

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).

Programs

Mathematica

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

PARI

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

Formula

a(n) = 1 if and only if n is in A090778.

a(n) = A000010(n) if and only if n is a cyclic number (A003277).

Let f(n) = a(n)/A373319(n). Then:

f(n) = A000010(n)/n^2 = A076512(n)/(n*A109395(n)).

f(n) = A173557(n)/A064549(n).

f(n) is multiplicative with f(p^e) = (1 - 1/p)/p^e.

Sum_{k=1..n} f(k) = (log(n) + gamma - zeta'(2)/zeta(2)) / zeta(2), where gamma is Euler's constant (A001620).