A373319 -id:A373319 - cp's OEIS frontend

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

Amiram Eldar, Jun 01 2024

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.

			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.

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

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).

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

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

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).

A373320 Numbers k such that phi(k)/k^2 < phi(m)/m^2 for all m < k, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 10, 12, 18, 24, 30, 42, 54, 60, 78, 84, 90, 114, 120, 150, 168, 180, 210, 270, 294, 300, 330, 390, 420, 510, 546, 570, 630, 750, 780, 840, 990, 1050, 1170, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3360, 3570, 3990, 4290, 4410, 4620

Offset: 1

Amiram Eldar, Jun 01 2024

nonn

First differs from A330006 at n = 52: a(52) = 4410 is not a term of A330006. The first term of A330006 that is not in this sequence is A330006(127) = 166530.
Numbers are less likely to be unitary divisors than any smaller number, i.e., numbers k such that the asymptotic density of numbers that are unitarily divided by k (A373318(k)/A373319(k)) is lower than the corresponding density of all m < k.
The numbers k such that phi(k)/k < phi(m)/m for all m < k are the primorial numbers (A002110).

Amiram Eldar, Table of n, a(n) for n = 1..670

Cf. A000010, A002110, A330006, A373318, A373319.

seq[kmax_] := Module[{rm = 2, r, s = {}}, Do[If[(r = EulerPhi[k]/k^2) < rm, rm = r; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[5000]

lista(kmax) = {my(rm = 2, r); for(k = 1, kmax, r = eulerphi(k)/k^2; if(r < rm, rm = r; print1(k, ", ")));}

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula