A082953 a(n) = A000252(n) / A070732(n).

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

Keywords

• mult
• nonn

Comments

From Jianing Song, Apr 20 2019: (Start)

a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].

a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000252, A065464, A070732.

Cf. A000010, A062570, A040001.

Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), 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

Maple

A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

Mathematica

Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)

PARI

a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

Formula

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005

Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011

a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019

Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 = 0.171299... . - Amiram Eldar, Oct 30 2022

a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025