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A053191 a(n) = n^2 * phi(n). 25
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 (list; graph; refs; listen; history; text; internal format)
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

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, 0, 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

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

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

Sequence in context: A049726 A130656 A292161 * A003474 A337003 A095823

Adjacent sequences:  A053188 A053189 A053190 * A053192 A053193 A053194

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

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Last modified November 26 09:38 EST 2020. Contains 338639 sequences. (Running on oeis4.)