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

1,2

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

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) = denominator(1/4) = 4.

Mathematica

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

Prog

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

Crossrefs

Cf. A003277, A373318 (numerators), A373320.

Sequence in context: A355012 A280286 A268597 * A253560 A050399 A072995

Adjacent sequences: A373313 A373317 A373318 * A373320 A373321 A373322

Keyword

nonn,easy,frac

Author

Amiram Eldar, Jun 01 2024

Status

approved