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

1,3

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.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Unitary Divisor.

Wikipedia, Unitary divisor.

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).

Example

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.

Mathematica

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

Prog

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

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).

Sequence in context: A187730 A049559 A063994 * A374127 A268336 A295127

Adjacent sequences: A373315 A373316 A373317 * A373319 A373320 A373321

Keyword

nonn,easy,frac

Author

Amiram Eldar, Jun 01 2024

Status

approved