OFFSET
1,2
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
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
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
EXAMPLE
n=5: n^3 = 125, EulerPhi(125) = 125 - 25 = 100.
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 *)
PROG
(Sage) [n^2*euler_phi(n) for n in range(1, 42)] # Zerinvary Lajos, Jun 06 2009
(Magma) [ n^2*EulerPhi(n) : n in [1..100] ]; // Vincenzo Librandi, Apr 21 2011
(PARI) a(n) = n^2*eulerphi(n); \\ Michel Marcus, Oct 31 2017
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Labos Elemer, Mar 02 2000
EXTENSIONS
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 05 2007
STATUS
approved