A053191 a(n) = n^2 * phi(n).

1, 4, 18, 32, 100, 72, 294, 256, 486, 400, 1210, 576, 2028, 1176, 1800, 2048, 4624, 1944, 6498, 3200, 5292, 4840, 11638, 4608, 12500, 8112, 13122, 9408, 23548, 7200, 28830, 16384, 21780, 18496, 29400, 15552, 49284, 25992, 36504, 25600, 67240

Offset: 1

Author

Labos Elemer, Mar 02 2000

Keywords

• nonn
• mult

Comments

Number of invertible 2 X 2 symmetric matrices over Z(n). - T. D. Noe, Jan 13 2006

Note that A115077 gives the number of 2 X 2 symmetric matrices having nonzero determinant. However, for composite n, a nonzero determinant is not sufficient for the matrix to be invertible; the determinant must also be relatively prime to n. - T. D. Noe, Jan 13 2006

Also Euler phi function of n^3.

For n^k, EulerPhi(n^k) = n^(k-1)*EulerPhi(n). The same holds if Phi is replaced by the cototient function.

Also, the sum of the degrees of the irreducible representations of the group GL(2,Z_n) (sequence A000252). - Sharon Sela (sharonsela(AT)hotmail.com), Feb 06 2002

Examples

			n=5: n^3 = 125, EulerPhi(125) = 125 - 25 = 100.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, A115076, A115077.

Cf. A000010, A051953, A002618, A053650, A053192, A001248, A319087.

Programs

Magma

[ n^2*EulerPhi(n) : n in [1..100] ]; // Vincenzo Librandi, Apr 21 2011

Maple

with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); # Zerinvary Lajos, Oct 07 2007

Mathematica

Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 2, 50}] (* T. D. Noe, Jan 13 2006 *)
Table[n^2*EulerPhi[n],{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Nov 10 2009 *)

PARI

a(n) = n^2*eulerphi(n); \\ Michel Marcus, Oct 31 2017

Sage

[n^2*euler_phi(n) for n in range(1, 42)] # Zerinvary Lajos, Jun 06 2009

Formula

a(n) = n^2 * phi(n) = A000010(n^3).

Dirichlet g.f.: zeta(s-3)/zeta(s-2). - R. J. Mathar, Feb 09 2011

The n-th term of the Dirichlet inverse is n^2 * A023900(n) = (-1)^omega(n) * a(n) / A003557(n), where omega = A001221. - Álvar Ibeas, Nov 24 2017

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^4 - p^3 - p + 1)) = 1.38097852211302096879... - Amiram Eldar, Dec 06 2020

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007