A127473 a(n) = phi(n)^2.

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 64, 64, 256, 36, 324, 64, 144, 100, 484, 64, 400, 144, 324, 144, 784, 64, 900, 256, 400, 256, 576, 144, 1296, 324, 576, 256, 1600, 144, 1764, 400, 576, 484, 2116, 256, 1764, 400, 1024, 576, 2704, 324, 1600

Offset: 1

Author

Gary W. Adamson, Jan 15 2007

Keywords

• nonn
• mult

Comments

Number of maps of the form j |--> m*j + d with gcd(m, n) = 1 and gcd(d, n) = 1 from [1, 2, ..., n] to itself. - Joerg Arndt, Aug 29 2014

Right border of A127474.

Equals the Mobius transform (A054525) of A029939. - Gary W. Adamson, Aug 20 2008

From Jianing Song, Apr 14 2019: (Start)

a(n) is the number of solutions to gcd(xy, n) = 1 with x, y in [0, n-1].

Let Z_n be the ring of integers modulo n, then a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - x) (or equivalently, Z_n[x]/(x^2 + x)) with discriminant d = 1 (that is, a(n) is the size of the group G(n) = (Z_n[x]/(x^2 - x))*). Actually, G(n) is isomorphic to (Z_n)* X (Z_n)*. (End)

Examples

			a(5) = 16 since phi(5) = 4.
		

Links

• Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Crossrefs

Cf. A000010, A057434, A065464, A109695, A127474, A358714, A361148, A008683.

Similar sequences: A082953 (size of (Z_n[x]/(x^2 - 1))*, d = 4), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

Programs

Magma

[(EulerPhi(n))^2: n in [1..180]]; // Vincenzo Librandi, Apr 04 2011

Maple

A127473 := proc(n) numtheory[phi](n)^2 ; end proc:
seq(A127473(n),n=1..40) ; # R. J. Mathar, Apr 04 2011

Mathematica

Table[EulerPhi[n]^2,{n,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

PARI

a(n) = eulerphi(n)^2; \\ Michel Marcus, Oct 16 2018

Formula

a(n) = A000010(n)^2.

Multiplicative with a(p^e) = (p-1)^2*p^(2e-2), e >= 1. Dirichlet g.f. zeta(s-2)*Product_{primes p} (1 - 2/p^(s-1) + 1/p^s). - R. J. Mathar, Apr 04 2011

Sum_{k>=1} 1/a(k) = A109695. - Vaclav Kotesovec, Sep 20 2020

Sum_{k>=1} (-1)^k/a(k) = (1/7) * A109695. - Amiram Eldar, Nov 11 2020

Sum_{k=1..n} a(k) ~ c * n^3, where c = (1/3) * Product_{p prime}(1 - (2*p-1)/p^3) = A065464 / 3 = 0.142749... . - Amiram Eldar, Oct 25 2022

a(n) = Sum_{d|n} mu(n/d)*phi(n*d). - Ridouane Oudra, Jul 23 2025