A090778 -id:A090778 - 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).

A275245 Numbers k such that phi(k) divides k^2 while phi(k) does not divide k.

Original entry on oeis.org

10, 20, 40, 42, 50, 60, 80, 84, 100, 114, 120, 126, 136, 156, 160, 168, 180, 200, 220, 228, 240, 250, 252, 272, 294, 300, 312, 320, 336, 342, 360, 378, 400, 440, 444, 456, 468, 480, 500, 504, 540, 544, 588, 600, 624, 640, 672, 684, 720, 756, 800, 816

Offset: 1

Altug Alkan, Jul 21 2016

nonn

			10 is a term because phi(10) = 4; 10 mod 4 = 2 and 10^2 mod 4 = 0.

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

Cf. A000010, A007694, A090778.

Select[Range[10^3], Function[k, And[Divisible[#^2, k], ! Divisible[#, k]]]@ EulerPhi@ # &] (* Michael De Vlieger, Jul 21 2016 *)

isok(n) = (n % eulerphi(n) != 0) && (n^2 % eulerphi(n) == 0)

cp's OEIS Frontend

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

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