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%I A053191 #51 Sep 08 2022 08:45:00
%S A053191 1,4,18,32,100,72,294,256,486,400,1210,576,2028,1176,1800,2048,4624,
%T A053191 1944,6498,3200,5292,4840,11638,4608,12500,8112,13122,9408,23548,7200,
%U A053191 28830,16384,21780,18496,29400,15552,49284,25992,36504,25600,67240
%N A053191 a(n) = n^2 * phi(n).
%C A053191 Number of invertible 2 X 2 symmetric matrices over Z(n). - _T. D. Noe_, Jan 13 2006
%C A053191 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
%C A053191 Also Euler phi function of n^3.
%C A053191 For n^k, EulerPhi(n^k) = n^(k-1)*EulerPhi(n). The same holds if Phi is replaced by the cototient function.
%C A053191 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
%H A053191 Seiichi Manyama, <a href="/A053191/b053191.txt">Table of n, a(n) for n = 1..10000</a>
%F A053191 a(n) = n^2 * phi(n) = A000010(n^3).
%F A053191 Dirichlet g.f.: zeta(s-3)/zeta(s-2). - _R. J. Mathar_, Feb 09 2011
%F A053191 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
%F A053191 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + p/(p^4 - p^3 - p + 1)) = 1.38097852211302096879... - _Amiram Eldar_, Dec 06 2020
%e A053191 n=5: n^3 = 125, EulerPhi(125) = 125 - 25 = 100.
%p A053191 with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); # _Zerinvary Lajos_, Oct 07 2007
%t A053191 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 *)
%t A053191 Table[n^2*EulerPhi[n],{n,1,40}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 10 2009 *)
%o A053191 (Sage) [n^2*euler_phi(n) for n in range(1, 42)] # _Zerinvary Lajos_, Jun 06 2009
%o A053191 (Magma) [ n^2*EulerPhi(n) : n in [1..100] ]; // _Vincenzo Librandi_, Apr 21 2011
%o A053191 (PARI) a(n) = n^2*eulerphi(n); \\ _Michel Marcus_, Oct 31 2017
%Y A053191 Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, A115076, A115077.
%Y A053191 Cf. A000010, A051953, A002618, A053650, A053192, A001248, A319087.
%K A053191 nonn,mult
%O A053191 1,2
%A A053191 _Labos Elemer_, Mar 02 2000
%E A053191 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 05 2007

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