A373318 -id:A373318 - cp's OEIS frontend

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

Amiram Eldar, Jun 01 2024

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 121, Erdős Problems.
Terence Tao, Erdős problem database, see no. 121.
Eric Weisstein's World of Mathematics, Unitary Divisor.
Wikipedia, Unitary divisor.

Cf. A003277, A373318 (numerators), A373320.

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

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

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

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

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

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

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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).

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